Topics On Stability And Periodicity In Abstract Differential Equations (Series on Concrete and Applicable Mathematics)

TOPICS ON STABILITY AND PERIODICITY IN ABSTRACT DIFFERENTIAL EQUATIONS SERIES ON CONCRETE AND APPLICABLE MATHEMATICS ...

Author: James H. Liu | Gaston M. N'Guerekata | Nguyen Van Minh

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TOPICS ON STABILITY AND PERIODICITY IN ABSTRACT DIFFERENTIAL EQUATIONS

SERIES ON CONCRETE AND APPLICABLE MATHEMATICS Series Editor: Professor George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA

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Series on Concrete and Applicable Mathematics – Vol. 6

TOPICS ON STABILITY AND PERIODICITY IN ABSTRACT DIFFERENTIAL EQUATIONS James H Liu James Madison University, USA

Gaston M N’Guérékata Morgan State University, USA

Nguyen Van Minh University of West Georgia, USA

World Scientific NEW JERSEY

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TOPICS ON STABILITY AND PERIODICITY IN ABSTRACT DIFFERENTIAL EQUATIONS Series on Concrete and Applicable Mathematics — Vol. 6 Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

Asymptotic behavior of evolution equations is a well-studied area in the theory of abstract differential equations with various methods of studies. It is natural to use the well-known ideas and techniques in the finite dimensional case as much as possible to deal with the problems in the infinite dimensional case. Having this in mind, in this book we will make an attempt to gather systematically certain recent results on several central topics of the asymptotic behavior of differential equations in Banach spaces. We will discuss the conditions for the stability, dichotomy and harmonic oscillation of solutions of evolution equations. The results and methods of approach will be presented in a manner that allows the reader, who is familiar with the techniques in the finite dimensional case, to easily understand them. Some parts of the book are actually lecture notes we have taught to graduate students over the past years. We outline briefly the contents of our book. In Chapter 1 we recall several basic facts from semigroup theory, spectral theory of functions that will be used throughout the book. Chapter 2 is devoted to some classical topics including stability and dichotomy of linear homogeneous equations. In Chapter 3 we present some new methods of studying the harmonic oscillation in inhomogeneous linear equations. Chapter 4 is devoted to the topic of almost automorphy of solutions, that has recently regained interest in the mathematical literature. Existence of almost automorphic solutions to some linear and semilinear abstract differential equations is studied. We discuss the Massera type conditions for the existence of periodic solutions to periodic nonlinear equations in Chapter 5. At the end of each chapter we give a guide for further reading and comments on the results as well as the methods of study discussed in the chapter. We finally collect some of the required tools from functional analysis and operator theory in the

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appendices. We wish to thank our colleagues and students for their encouragement and patience during the last years. James H. Liu, Gaston M. N’guerekata, Nguyen Van Minh

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Contents

Preface

v

1. Preliminaries

1

1.1

1.2

1.3

Banach Spaces and Linear Operators . . . . . . . . . . . . 1.1.1 Banach Spaces . . . . . . . . . . . . . . . . . . . . 1.1.2 Linear Operators . . . . . . . . . . . . . . . . . . 1.1.3 Spectral Theory of Linear (Closed) Operators . . Strongly Continuous Semigroups of Operators . . . . . . . 1.2.1 Definition and Basic Properties . . . . . . . . . . 1.2.2 Compact Semigroups and Analytic Strongly Continuous Semigroups . . . . . . . . . . . . . . . . . 1.2.3 Spectral Mapping Theorems . . . . . . . . . . . . 1.2.4 Commuting Operators . . . . . . . . . . . . . . . Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1.3.2 Spectrum of a Bounded Function . . . . . . . . . 1.3.3 Uniform Spectrum of a Bounded Function . . . . 1.3.4 Almost Periodic Functions . . . . . . . . . . . . . 1.3.5 Sprectrum of an Almost Periodic Function . . . . 1.3.6 A Spectral Criterion for Almost Periodicity of a Function . . . . . . . . . . . . . . . . . . . . . . . 1.3.7 Almost Automorphic Functions . . . . . . . . . .

2. Stability and Exponential Dichotomy 2.1 2.2

Perron Theorem . . . . . . . . . . . . . . . . . . . . . . . Evolution Semigroups and Perron Theorem . . . . . . . . vii

1 1 2 3 7 7 12 14 17 19 19 19 24 26 29 30 31 39 39 47

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2.3

2.4

Stability Theory . . . . . . . . . . . . 2.3.1 Exponential Stability . . . . . 2.3.2 Strong Stability . . . . . . . . Comments and Further Reading Guide 2.4.1 Further Reading Guide . . . . 2.4.2 Comments . . . . . . . . . . .

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3. Almost Periodic Solutions 3.1

3.2

3.3

3.4

Evolution Semigroups & Periodic Equations . . . . . . . . 3.1.1 An Example . . . . . . . . . . . . . . . . . . . . . 3.1.2 Evolution Semigroups . . . . . . . . . . . . . . . . 3.1.3 The Finite Dimensional Case . . . . . . . . . . . . 3.1.4 The Infinite Demensional Case . . . . . . . . . . . 3.1.5 Almost Periodic Solutions and Applications . . . Sums of Commuting operators . . . . . . . . . . . . . . . 3.2.1 Invariant Function Spaces . . . . . . . . . . . . . 3.2.2 Differential Operator d/dt − A and Notions of Admissibility . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Admissibility for Abstract Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Higher Order Differential Equations . . . . . . . . 3.2.5 Abstract Functional Differential Equations . . . . 3.2.6 Examples and Applications . . . . . . . . . . . . . Decomposition Theorem . . . . . . . . . . . . . . . . . . 3.3.1 Spectral Decomposition . . . . . . . . . . . . . . . 3.3.2 Spectral Criteria For Almost Periodic Solutions . Comments and Further Reading Guide . . . . . . . . . . . 3.4.1 Further Reading Guide . . . . . . . . . . . . . . . 3.4.2 Comments . . . . . . . . . . . . . . . . . . . . . .

4. Almost Automorphic Solutions 4.1 4.2 4.3 4.4

The Inhomogeneous Linear Equation . . . . . . . . . . . . Method of Invariant Subspaces and Almost Automorphic Solutions of Second-Order Differential Equations . . . . . Existence of Almost Automorphic Solutions to Semilinear Differential Equations . . . . . . . . . . . . . . . . . . . . Method of Sums of Commuting Operators and Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . .

51 51 54 58 58 59 61 61 61 63 64 65 68 81 81 83 86 89 96 98 103 107 114 118 118 119 121 121 127 131 135

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Contents

4.5

4.6 4.7

ix

Almost Automorphic Solutions of Second Order Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Mild Solutions of Inhomogeneous Second Order Equations . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Operators A . . . . . . . . . . . . . . . . . . . . . 4.5.3 Nonlinear Equations . . . . . . . . . . . . . . . . . The Equations x’=f(t,x) . . . . . . . . . . . . . . . . . . . Comments and Further Reading Guide . . . . . . . . . . .

5. Nonlinear equations 5.1

5.2

5.3

Appendix A.1 A.2 A.3 A.4

139 140 141 145 146 151 153

Periodic Solutions of Nonlinear equations . . . . . . 5.1.1 Nonlinear Equations Without Delay . . . . . 5.1.2 Nonlinear Equations With Finite Delay . . . 5.1.3 Nonlinear Equations With Infinite Delay . . 5.1.4 Non-Densely Defined Equations . . . . . . . Evolution Semigroups and Almost Periodic Solutions 5.2.1 Evolution Semigroups . . . . . . . . . . . . . 5.2.2 Almost periodic solutions . . . . . . . . . . . Comments and Further Reading Guide . . . . . . . . 5.3.1 Further Reading Guide . . . . . . . . . . . . 5.3.2 Comments . . . . . . . . . . . . . . . . . . .

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153 153 162 166 180 183 183 186 190 190 191

Lipschitz Operators . . . . . . Fixed Point Theorems . . . . . Invariant Subspaces . . . . . . Semilinear Evolution Equations

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193 193 195 197 198

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Bibliography

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Index

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Chapter 1

Banach Spaces, C0-Semigroups of Linear Operators and Almost Periodicity of Functions 1.1 1.1.1

Banach Spaces and Linear Operators Banach Spaces

The notion of Banach spaces will be used throughout this book. A normed space is a linear space X, endowed with a norm, frequently denoted by k · k, that is a function from X to the set of all real numbers, denoted by R, such that (1) kxk ≥ 0, kxk = 0 if and only if x = 0; (2) kλxk = |λ|kxk, ∀λ ∈ C or R, x ∈ X; (3) kx + yk ≤ kxk + kyk, ∀ x, y ∈ X. A normed space X is called Banach space if it is complete, i.e., every Cauchy sequence in X is convergent. Example 1.1. Let BC(R, X) be the linear space of all bounded continuous X-valued functions on R with sup-norm kf k := sup kf (t)k, ∀f ∈ BC(R, X).

(1.1)

t∈R

Then BC(R, X) is a Banach space. Similarly, the following spaces are Banach spaces if endowed with norm (1.1). BU C(R, X) := {f ∈ BC(R, X) : f is uniformly continuous}

(1.2)

Pω := {f ∈ BC(R, X) : f is periodic with period ω} (1.3)

However, the function space C 1 (R, X) := {f ∈ BC(R, X) : f 0 exists and f 0 ∈ BC(R, X)} 1

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2

is not a Banach space. In fact, it is easy to choose a sequence of differentiable functions {fn } with fn0 ∈ BC(R, X) such that it is a Cauchy sequence but it does not converge to a differentiable function. Example 1.2. Let Ω be the unit open ball of Rn , i.e., Ω = {x ∈ Rn : kxk < 1}. We denote by C m (Ω) the set of all m times continuously differentiable functions in Ω with the derivatives up to the order m bounded and continuously extendable up to the boundary {x ∈ Rn : kxk = 1}. Then C m (Ω) is a Banach space with the following norm X sup kDα f (x)k. (1.4) kf kC m (Ω) := |α|≤m

x∈Ω

To conclude this subsection we consider the following Banach spaces Example 1.3. For any interval I = [a, b], a < b ∈ R and α ∈ (0, 1) let us denote kf (t) − f (s)k C α (I, X) := {f ∈ BC(I, X) : sup < ∞}. (t − s)α t,s∈I,s
Then C α (I, X) is a Banach space with the norm kf (t) − f (s)k kf kC α(I,X) := sup kf (t)k + sup . (t − s)α t∈I t,s∈I,s
These Banach spaces are called the Banach spaces of H¨ older continuous functions. In general if dimX < ∞, then X is a Banach space with any norm. Exercise 1. Show that X with norm k·k is a finite-dimensional Banach space if and only if the unit ball {x ∈ X : kxk ≤ 1} is compact.

1.1.2

Linear Operators

Definition 1.1. Let X be a Banach space. Then a mapping A from D(A) ⊂ X to X is said to be a linear operator if D(A) is a linear subspace of X and A is linear. In this case D(A) is called domain of A and the range of this operator will be denoted by R(A). Remark 1.1. In the definition of a linear operator the domain is necessarily a linear space. In general, it is a dense subspace of X but not the whole space X.

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3

Example 1.4. Let M be an n × n matrix with real entries. Then it defines a linear operator from Rn into itself by the rule x 7→ M x, where x is a columm of n rows, an element of Rn . In this example, denoting by M the corresponding linear operator, we have that D(M) = Rn . Generally, if dimX < ∞, then from the density of D(A) in X follows that D(A) = X. However, it is not the case for the following operators in infinite dimensional Banach spaces: Example 1.5. Let A be the differential operator d/dt with D(A) defined as follows: D(A) = {f ∈ BC(R, X) : df /dt ∈ BC(R, X)}. Obviously, A is a linear operator and D(A) is a subspace of X that is dense everywhere in X, but is not X. Example 1.6. Let B be the differential operator d/dt with D(B) defined as follows: D(B) := {g ∈ BC(R, X) : dg/dt ∈ BC(R, X), dg(0)/dt = 1}. It is not difficult to see that B is not a linear subspace of BC(R, X) as the domain D(B) is not linear subspace. In fact, we can see that 0 is not in D(B). 1.1.3

Spectral Theory of Linear (Closed) Operators

First, we introduce the notion of bounded linear operators on a Banach space X, and then extend our consideration to more general classes of linear operators, for instance, closed operators. Definition 1.2. Let A be a linear operator on a Banach space X with D(A) = X. Then it is said to be a bounded linear operator on X if there exists a positive constant c such that kAxk ≤ ckxk, ∀x ∈ X.

(1.5)

Hence, if A is a bounded linear operator, then it is continuous. Exercise 2. Show the converse of the above assertion. In view of this exercise, the notion of bounded linear operators is nothing but that of continuous linear operators.

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Let A be a bounded linear operator. Then the following nonnegative number kAk := inf {c ∈ R : kAxk ≤ ckxk, ∀x ∈ X}

(1.6)

is called the norm of A. Exercise 3. Let L(X) denote the set of all bounded linear operators on the given Banach space X. Then L(X) endowed with the norm (1.6) is again a Banach space. As we have seen in the above example, there are linear operators that are not bounded. Among the class of unbounded linear operators the class of closed operators is particularly important. There are two reasons, to our view, for this importance. The first one is that we encounter the operators of this class everywhere in problems involving partial differential equations, functional differential equations, integro-differential equations, etc. The second one is that the requirement on the closedness is indeed not too much. Every linear operator with nonempty resolvent set is closed, as shown below. Now we give a precise definition of this class. Definition 1.3. A linear operator A from D(A) ⊂ X to X is said to be closed if its graph, i.e., the set {(x, Ax) ∈ X × X, ∀x ∈ D(A)} is closed. If a linear operator A is not closed, one may expect that there is an extension so that the extension of it is closed. In this case, we say that the linear operator A is closable. The smallest extension is called the closure of A. We are ready to define the notion of spectrum of a closed linear operator A. Let X be a given complex Banach space. Definition 1.4. We call the set ρ(A) := {λ ∈ C : λ − A : D(A) → X is bijective}

the resolvent set and its complement σ(A) := C\ρ(A) the spectrum of A. For λ ∈ ρ(A), the inverse R(λ, A) := (λ − A)−1

is, by the closed graph theorem, a bounded linear operator on X and will be called the resolvent of A in the point λ. Remark 1.2. Definition (1.4) can be extended to general linear operators, which are not necessarily closed, if we require that the map A : D(A) ⊂ X → X has bounded inverse.

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5

Exercise 4. Show that if a linear operator A : D(A) ⊂ X → X has nonempty resolvent set, then it is closed. Example 1.7. Let A be any linear operator of the finite dimensional complex Banach space Cn . Then σ(A) is exactly the set of all eigenvalues of A, i.e. the set of all λ ∈ C such that there is a nonzero vector x of Cn with Ax = λx. Any bounded linear operator A has nonempty spectrum. Its spectrum is contained in the disk of radius 1/n . (1.7) r(A) := lim kAkn n→∞

This number is called spectral radius of the bounded operator A. In contrast to the finite dimensional case, in general, a bounded linear spectrum may have no eigenvalue, for instance,

Example 1.8. Let c0 be the Banach space of numerical two-sided sequences which converge to zero with sup-norm, i.e., c0 := {{xn }∞ n=0 , xn ∈ R, lim xn = 0}. n→∞

Then we define the translation T : c0 → c0 which maps every sequence {xn } to the sequence {yn } such that yn = xn−1 , ∀n > 1, y1 = 0. It is seen that ∞ \ T n c0 = {0}. n=1

So, if there is an eigenvalue λ, then there is an invariant nontrivial subspace of T . This is impossible. 1.1.3.1 Several Properties of Resolvents An important property of ρ(A) is that it is an open subset of the complex plane. The map ρ(A) 3 λ → R(λ, A) ∈ L(X) (resolvent map) is analytic in the open subset ρ(A). One can show that any closed subset of the complex plane can serve as a spectrum of a closed linear operator. Theorem 1.1. For a closed linear operator A : D(A) ⊂ X → X, the following properties hold true:

(1) The resolvent set ρ(A) is open in C, and for µ ∈ ρ(A) one has ∞ X R(λ, A) = (µ − λ)n R(µ, A)n+1 n=0

for all λ ∈ C such that |µ − λ| < 1/kR(µ, A)k.

(1.8)

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(2) The resolvent map λ 7→ R(λ, A) is locally analytic with

dn R(λ, A) = (−1)n n!R(λ, A)n+1 , ∀n ∈ N. dλn

(3) Let λn ∈ ρ(A) with limit limn→∞ λn = λ0 . Then λ0 ∈ σ(A) if and only if lim kR(λn , A)k = ∞.

n→∞

Proof.

For the proof see e.g. Chap. IV in [Engel and Nagel (30)].

The following is concerned with another elementary property of resolvents: Theorem 1.2. Let A be a closed linear operator on a Banach space X. Then for all λ ∈ ρ(A) we have kR(λ, A)k ≥

1 . dist (λ, σ(A))

Proof. This will be an immediate consequence of the fact that kR(λ, A)k ≥ r(R(λ, A)) once we prove that σ(R(λ, A)) =

1 . λ − σ(A)

But it is trivial to check that (λ − µ)(λ − A)R(µ, A) is a two-sided inverse for (λ − µ)−1 − R(λ, A) whenever µ ∈ ρ(A), which proves the inclusion ⊂. Similarly, (λ − µ)−1 R(λ, A)((λ − µ)−1 − R(λ, A))−1 is a two-sided inverse for µ − A whenever (λ − µ)−1 ∈ ρ(R(λ, A)), which proves the inclusion ⊃. Let A be a closed linear operator. Then we introduce some finer notions of spectrum for A. Definition 1.5. (1) Each λ ∈ σ(A) such that λ − A is not injective, is called eigenvalue of A, and each nonzero vector x ∈ D(A) such that (λ − A)x = 0 is called eigenvector corresponding to λ. The subset of σ(A) consisting of all eigenvalues of A, is denoted by P σ(A) and is called the point spectrum of A. (2) We call approximate eigenvalue each λ ∈ σ(A) such that there is a sequence {xn } ⊂ D(A) ( called approximate eigenvector) satisfying kxn k = 1 and limn→∞ kAxn − λxn k = 0.

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1.2

7

Strongly Continuous Semigroups of Operators

In this section we collect some well-known facts from the theory of strongly continuous semigroups of operators on a Banach space for the reader’s convenience. We will focus the reader’s attention on several important classes of semigroups such as analytic and compact semigroups which will be discussed in the next chapters. Among the basic properties of strongly continuous semigroups we will emphasize on the spectral mapping theorem. Since the materials of this section as well as of this chapter in the whole can be found in any standard book covering the area, here we aim at freshening up the reader’s memory rather than giving a logically self contained account of the theory. 1.2.1

Definition and Basic Properties

Definition 1.6. A family (T (t))t≥0 of bounded linear operators acting on a Banach space X is called a C0 -semigroup if the following three properties are satisfied: (1) T (0) = I, the identity operator on X; (2) T (t)T (s) = T (t + s) for all t, s ≥ 0; (3) limt→0+ kT (t)x − xk = 0 for all x ∈ X. The infinitesimal generator of (T (t))t≥0 , or briefly the generator, is the linear operator A with domain D(A) defined by 1 D(A) = {x ∈ X : lim (T (t)x − x) exists}, t↓0 t 1 Ax = lim (T (t)x − x), x ∈ D(A). t↓0 t The generator is always a closed, densely defined operator. A strongly continuous semigroup of bounded linear operators on X will be called C0 semigroup. We now consider several examples of strongly continuous semigroups. Example 1.9. Let A be a bounded linear operator on a Banach space X. Then (etA ))t≥0 , defined by the formula etA :=

∞ X (tA)k

k=0

k!

,

(1.9)

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is a strongly continuous semigroup of bounded linear operators on the Banach space X. Moreover, its generator is the operator A with D(A) = X. Proof. First, it may be noted that the formula (1.9) is well defined. In fact, since A is bounded ∞ X k(tA)k k ≤ e|t|kAk , ∀t. k! k=0 Hence, the series is absolutely convergent. To check that this family is indeed a semigroup we will prove the following e(t+s)A = etA esA , ∀s, t ∈ R. From the absolute convergence of the above series it follows that the product n k P∞ P∞ sA := n=0 (sA) is absolutely convergent, of two series k=0 (tA) k! , and e n! i.e., ∞ ∞ X (tA)k X (sA)n × k! n! n=0 k=0 is convergent. Moreover, it does not depend on the way of summation, in particular, ∞ ∞ X (tA)k X (sA)n etA esA = × k! n! n=0 = =

k=0 ∞ X

X

m=0 k+n=m ∞ m X

(tA)k (sA)n k! n!

(tA) m! m=0

(1.10) (1.11)

= e(t+s)A . (1.12) We now show that this semigroup is strongly continuous. By definition, we have to show that lim etA x = x, ∀x ∈ X. By (1.9), ∀x ∈ X

t→0+

ketA x − xk ≤ k

∞ X (tAx)k x k k! k=1

≤ |t|kAkkxk ≤ |t|kAkkxk

∞ X k(tA)k k k=0 ∞ X

k=0

(k + 1)!

k(tAx)k k (k)!

= |t|kAkkxke|t|kAk.

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9

Hence lim ketA x − xk = 0.

t→0+

Now we show that A is the generator of this semigroup with D(A) = X, i.e., we have to show that for all x ∈ X, the following holds true lim

t→0+

etA x − x = Ax. t

(1.13)

In fact, by (1.9), ∀x ∈ X

etA x − x − Axk ≤ |t|e|t|kAk kAkkxk. t Hence (1.13) holds true. k

(1.14)

Example 1.10. Let (S(t))t≥0 be the translation semigroup on BU C(R, X), where X is a Banach space, i.e., S(t)f (s) := f (t + s), ∀t ≥ 0, s ∈ R, f ∈ BU C(R, X). It is easy to check that (S(t))t≥0 is a semigroup of bounded linear operators on BU C(R, X). From the uniform continuity of every function in BU C(R, X) it follows that this translation semigroup is strongly continuous. As an example of a non strongly continuous semigroup we can consider the translation semigroup (S(t))t≥0 in BC(R, X). The non-strong continuity follows from the fact that there exists a bounded function which is not uniformly continuous. Indeed, we can take the following example Example 1.11. The function f (t) = sin t2 , t ∈ R, is a continuous and bounded function that is not uniformly continuous. p p In fact, we can choose a sequence t2n = 2nπ + π/2, t2n+1 = (2n + 1)π. Obviously, f (t−2n) = 1 and f (t2n+1 ) = 0 while |t2n −t2n+1 | → 0 as n → ∞. Theorem 1.3. Let (T (t))t≥0 be a C0 -semigroup. Then there exist constants ω ≥ 0 and M ≥ 1 such that kT (t)k ≤ M eωt , ∀t ≥ 0.

Proof.

For the proof see e.g. p. 4 in [Pazy (90)].

Corollary 1.1. If (T (t))t≥0 is a C0 -semigroup, then the mapping (x, t) 7→ T (t)x is a continuous function from X × R+ → X.

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For any x, y ∈ X and t ≤ s ∈ R+ := [0, ∞),

kT (t)x − T (s)yk ≤ kT (t)x − T (s)xk + kT (s)x − T (s)yk

≤ M eωs kx − yk + kT (t)kkT (s − t)x − xk ≤ M eωs kx − yk + M eωt kT (s − t)x − xk.

(1.15)

Hence, for fixed x, t (t ≤ s) if (y, s) → (x, t) then kT (t)x − T (s)yk → 0. Similarly, for s ≤ t kT (t)x − T (s)yk ≤ kT (t)x − T (s)xk + kT (s)x − T (s)yk

≤ M eωs kx − yk + kT (s)kkT (t − s)x − xk ≤ M eωs kx − yk + M eωs kT (t − s)x − xk.

Hence, if (y, s) → (x, t) then kT (t)x − T (s)yk → 0.

(1.16)

Other basic properties of a C0 -semigroup and its generator are listed in the following: Theorem 1.4. Let A be the generator of a C0 -semigroup (T (t))t≥0 on X. Then (1) For x ∈ X,

(2) For x ∈ X,

1 lim h→0 h Rt 0

Z

t+h

T (s)xds = T (t)x. t

T (s)xds ∈ D(A) and Z t T (s)xds = T (t)x − x. A 0

(3) For x ∈ D(A), T (t)x ∈ D(A) and d T (t)x = AT (t)x = T (t)Ax. dt (4) For x ∈ D(A), Z t Z t T (t)x − T (s)x = T (τ )Axdτ = AT (τ )xdτ. s

Proof.

s

For the proof see e.g. p. 5 in [Pazy (90)].

We continue with some useful facts about semigroups that will be used throughout this book. The first of these is the Hille-Yosida theorem that characterizes the generators of C0 -semigroups among the class of all linear operators. Theorem 1.5. Let A be a linear operator on a Banach space X, and let ω ∈ R and M ≥ 1 be constants. Then the following assertions are equivalent:

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(1) A is the generator of a C0 -semigroup (T (t))t≥0 satisfying kT (t)k ≤ M eωt for all t ≥ 0; (2) A is closed, densely defined, the half line (ω, ∞) is contained in the resolvent set ρ(A) of A, and we have the estimates kR(λ, A)n k ≤

M , (λ − ω)n

∀λ > ω,

n = 1, 2, ...

(1.17)

Here, R(λ, A) := (λ−A)−1 denotes the resolvent of A at λ. If one of the equivalent assertions of the theorem holds, then actually {Reλ > ω} ⊂ ρ(A) and kR(λ, A)n k ≤

M , (Reλ − ω)n

∀Reλ > ω,

n = 1, 2, ...

Moreover, for Reλ > ω the resolvent is given explicitly by Z ∞ R(λ, A)x = e−λt T (t)x dt, ∀x ∈ X.

(1.18)

(1.19)

0

We shall mostly need the implication (i)⇒(ii), which is the easy part of the theorem. In fact, one checks directly from the definitions that Z ∞ Rλ x := e−λt T (t)x dt 0

defines a two-sided inverse for λ − A. The estimate (1.18) and the identity (1.19) follow trivially from this. A useful consequence of (1.17) is that lim kλR(λ, A)x − xk = 0,

λ→∞

∀x ∈ X.

(1.20)

This is proved as follows. Fix x ∈ D(A) and µ ∈ ρ(A), and let y ∈ X be such that x = R(µ, A)y. By (1.17) we have kR(λ, A)k = O(λ−1 ) as λ → ∞. Therefore, the resolvent identity R(λ, A) − R(µ, A) = (µ − λ)R(λ, A)R(µ, A)

(1.21)

implies that lim kλR(λ, A)x − xk = lim kR(λ, A)(µR(µ, A)y − y)k = 0.

λ→∞

λ→∞

This proves (1.20) for elements x ∈ D(A). Since D(A) is dense in X and the operators λR(λ, A) are uniformly bounded as λ → ∞ by (1.17), (1.20) holds for all x ∈ X.

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Topics on Stability and Periodicity in Abstract Differential Equations

1.2.2

Compact Semigroups and Analytic Strongly Continuous Semigroups

stability

Definition 1.7. A C0 -semigroup (T (t))t≥0 is called compact for t > t0 if for every t > t0 , T (t) is a compact operator. (T (t))t≥0 is called compact if it is compact for each t > 0. If a C0 -semigroup (T (t))t≥0 is compact for t > t0 , then it is continuous in the uniform operator topology for t > t0 . Theorem 1.6. Let A be the generator of a C0 -semigroup (T (t))t≥0 . Then (T (t))t≥0 is a compact semigroup if and only if T (t) is continuous in the uniform operator topology for t > 0 and R(λ; A) is compact for λ ∈ ρ(A). Proof.

For the proof see e.g. p. 49 in [Pazy (90)].

In this book we distinguish the notion of analytic C0 -semigroups from that of analytic semigroups in general. To this end we recall several notions. Let A be a linear operator D(A) ⊂ X → X with not necessarily dense domain. Definition 1.8. A is said to be sectorial if there are constants ω ∈ R, θ ∈ (π/2, π), M > 0 such that the following conditions are satisfied:    i) ρ(A) ⊃ Sθ,ω = {λ ∈ C : λ 6= ω, |arg(λ − ω)| < θ},   ii) kR(λ, A)k ≤ M/|λ − ω| ∀λ ∈ S . θ,ω

If we assume in addtion that ρ(A) 6= , then A is closed. Thus, D(A), endowed with the graph norm kxkD(A) := kxk + kAxk,

is a Banach space. For a sectorial operator A, from the definition, we can define a linear bounded operator etA by means of the Dunford integral Z 1 tA etλ R(λ, A)dλ, t > 0, (1.22) e := 2πi ω+γr,η

where r > 0, η ∈ (π/2, θ) and γr,η is the curve

{λ ∈ C : |argλ| = η, |λ| ≥ rk} ∪ {λ ∈ C : |argλ| ≤ η, |λ| = r},

oriented counterclockwise. In addition, set e0A x = x, ∀x ∈ X.

Theorem 1.7. Under the above notation, for a sectorial operator A the following assertions hold true:

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(1) etA x ∈ D(Ak ) for every t > 0, x ∈ X, k ∈ N. If x ∈ D(Ak ), then Ak etA x = etA Ak x, ∀t ≥ 0; (2) etA esA = e(t+s)A , ∀t, s ≥ 0; (3) There are positive constants M0 , M1 , M2 , ..., such that  tA ωt   (a) ke k ≤ M0 e , t ≥ 0,

  (b) ktk (A − ωI)k etA k ≤ M eωt , t ≥ 0, k

where ω is determined from Definition 1.8. In particular, for every ε > 0 and k ∈ N there is Ck,ε such that ktk Ak etA k ≤ Ck,ε e(ω+ε)t , t > 0; (4) The function t 7→ etA belongs to C ∞ ((0, +∞), L(X)), and dk tA e = Ak etA , t > 0, dtk moreover it has an analytic extension in the sector S = {λ ∈ C : |argλ| < θ − π/2}. Proof.

For the proof see pp. 35-37 in [Lunardi (65)].

Definition 1.9. For every sectorial operator A the semigroup (etA )t≥0 defined in Theorem 1.7 is called the analytic semigroup generated by A in X. An analytic semigroup is said to be an analytic strongly continuous semigroup if in addition, it is strongly continuous. There are analytic semigroups which are not strongly continuous, for instance, the analytic semigroups generated by nondensely defined sectorial operators. From the definition of sectorial operators it is obvious that for a sectorial operator A the intersection of the spectrum σ(A) with the imaginary axis is bounded. In this book, if otherwise stated, by ”analytic semigroups” we mean analytic semigroups that are strongly continuous. We use this convention because most of the results presented here are concerned with C0 -semigroups.

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Topics on Stability and Periodicity in Abstract Differential Equations

1.2.3

Spectral Mapping Theorems

If A is a bounded linear operator on a Banach space X, then by the Dunford Theorem [Dunford and Schwartz (29)] σ(exp(tA)) = exp(tσ(A)), ∀t ≥ 0. It is natural to expect this relation holds for any C0 -semigroups on a Banach space. However, this is not true in general as shown by the following counterexample (p. 44 in [Pazy (90)]) Example 1.12. Let X be the Banach space of all continuous functions on the interval [0, 1] which are equal to zero at x = 1 with the supremum norm. Define ( f (x + t), if x + t ≤ 1 (T (t)f )(x) = 0, if x + t > 1. (T (t))t≥0 is obviously a C0 -semigroup of contraction on X. Its generator A is given by D(A) = {f : f ∈ C 1 ([0, 1]) ∩ X, f 0 ∈ X} and Af = f 0 ,

for f ∈ D(A).

For every λ ∈ C and g ∈ X the equation λf − f 0 = g has a unique solution f ∈ X given by Z 1 f (t) = eλ(t−s) g(s)ds. t

Therefore, σ(A) = ∅. On the other hand, since for every t ≥ 0, T (t) is a bounded operator, σ(T (t)) 6= 0, so the relation σ(T (t)) = exp(tσ(A)) does not hold for any t ≥ 0.

In this section we prove the following Spectral Inclusion Theorem for C0 -semigroups: Theorem 1.8. Let (T (t))t≥0 be a C0 -semigroup on a Banach space X with generator A. Then we have the spectral inclusion relation σ(T (t)) ⊃ etσ(A) ,

∀t ≥ 0.

Proof. By Theorem 1.4 for the semigroup (T λ (t))t≥0 := {e−λt T (t)}t≥0 generated by A − λ, for all λ ∈ C and t ≥ 0 Z t (λ − A) eλ(t−s) T (s)x ds = (eλt − T (t))x, ∀x ∈ X, 0

stability

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15

and Z

t 0

eλ(t−s) T (s)(λ − A)x ds = (eλt − T (t))x,

∀x ∈ D(A).

(2.1.1)

Suppose eλt ∈ ρ(T (t)) for some λ ∈ C and t ≥ 0, and denote the inverse of eλt − T (t) by Qλ,t . Since Qλ,t commutes with T (t) and hence also with A, we have Z t (λ − A) eλ(t−s) T (s)Qλ,t x ds = x, ∀x ∈ X, 0

and

Z

t 0

eλ(t−s) T (s)Qλ,t (λ − A)x ds = x,

∀x ∈ D(A).

This shows the boundedness of the operator Bλ defined by Z t eλ(t−s) T (s)Qλ,t x ds Bλ x := 0

is a two-sided inverse of λ − A. It follows that λ ∈ %(A).

As shown by Example 1.12 the converse inclusion etσ(A) ⊃ σ(T (t))\{0} in general fails. For certain parts of the spectrum, however, the Spectral Mapping Theorem holds true. To make it more clear we recall that for a given closed operator A on a Banach space X the point spectrum σp (A) is the set of all λ ∈ σ(A) for which there exists a non-zero vector x ∈ D(A) such that Ax = λx, or equivalently, for which the operator λ − A is not injective; the residual spectrum σr (A) is the set of all λ ∈ σ(A) for which λ − A does not have dense range; the approximate point spectrum σa (A) is the set of all λ ∈ σ(A) for which there exists a sequence (xn ) of norm one vectors in X, xn ∈ D(A) for all n, such that lim kAxn − λxn k = 0.

n→∞

Obviously, σp (A) ⊂ σa (A). Theorem 1.9. Let (T (t))t≥0 be a C0 -semigroup on a Banach space X, with generator A. Then σp (T (t))\{0} = etσp (A) , Proof.

∀t ≥ 0.

For the proof see e.g. p. 46 in [Pazy (90)].

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16

Recall that a family of bounded linear operators (T (t))t∈R is said to be a strongly continuous group if it satisfies (1) T (0) = I, (2) T (t + s) = T (t)T (s), ∀t, s ∈ R, (3) limt→0 T (t)x = x, ∀x ∈ X. Similarly as C0 -semigroups, the generator of a strongly continuous group (T (t))t∈R is defined to be the operator Ax := lim

t→0

T (t)x − x , t

with the domain D(A) consisting of all elements x ∈ X such that the above limit exists. In the next chapter we need the following lemma: Lemma 1.1. Let (T (t))t≥0 be a uniformly bounded C0 -group on a Banach space X 6= {0}, with generator A. Then σ(A) 6= ∅. For bounded strongly continuous groups of linear operators the following Weak Spectral Mapping Theorem holds: Theorem 1.10. Let (T (t))t∈R be a bounded strongly continuous group, i.e., there exists a positive M such that kT (t)k ≤ M, ∀t ∈ R with generator A. Then σ(T (t)) = etσ(A) , ∀t ∈ R. Proof. (78)].

(1.23)

For the proof see e.g. [Nagel (73)] or Chapter 2 in [van Neerven

Example 1.13. Let M be a closed translation invariant subspace of the space of X-valued bounded uniformly continuous functions on the real line BU C(R, X), i.e., M is closed and S(t)M ⊂ M, ∀t, where (S(t))t∈R is the translation group on BU C(R, X). Then σ(S(t)|M ) = etσ(DM ) , ∀t ∈ R, where DM is the generator of the restriction of translation group to M. In the next chapter we will consider situations similar to this example which arise in connection with invariant subspaces of so-called evolution semigroups.

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1.2.4

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17

Commuting Operators

Let A, B be two bounded linear operators on a given Banach space X which is assumed to be complex. The definition of commutativeness of these operators in this case is natural, i.e., the identity AB = BA holds. In the general case, where the operators A and B are not necessarily everywhere defined, we have the following definition: Definition 1.10. Let A and B be linear operators on a Banach space G with non-empty resolvent sets. We say that A and B commute if one of the following equivalent conditions hold: (1) R(λ, A)R(µ, B) = R(µ, B)R(λ, A) for some (all) λ ∈ ρ(A), µ ∈ ρ(B), (2) x ∈ D(A) implies R(µ, B)x ∈ D(A) and AR(µ, B)x = R(µ, B)Ax for some (all) µ ∈ ρ(B). Exercise 5. Show that if A, B are bounded linear operators which are commutative in the usual sense, i.e., AB = BA, then they are commuting operators in the sense of Definition 1.10. Exercise 6. Show that if A is a bounded linear operator on a Banach space X and (T (t))t≥0 is a strongly continuous semigroup on X. Then, if A commutes with T (t) for all t ≥ 0, A must commute with the infinitesimal generator of the semigroup (T (t))t≥0 . Let us consider the generator of the product of two commuting semigroups (G(t))t≥0 , (H(t))t≥0 , that is, the semigroup (P (t))t≥0 defined by P (t) = G(t) · H(t). Generally, this semigroup may not be strongly continuous. Below, we assume that the product semigroup (P (t))t≥0 is strongly continuous. Let us denote by G, H, P the generators of (G(t))t≥0 , (H(t))t≥0 , (P (t))t≥0 , respectively. Exercise 7. Under the above assumptions and notations prove that P = G + H. Hint. First show that for every t ≥ 0, G(t)D(H) ⊂ D(H) from which the following holds: D(G) ∩ D(H) ⊂ D(P). Next, for every x ∈ X, t > 0 using the following Z Z t 1 t y(t) = 2 H(ξ) G(η)xdξdη (1.24) t 0 0 to prove that D(G) ∩ D(H) is dense in X. Using this fact, complete the proof by showing that G + H = P.

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For θ ∈ (0, π), R > 0 we denote Σ(θ, R) = {z ∈ C : |z| ≥ R, |argz| ≤ θ}. Definition 1.11. Let A and B be commuting operators. Then (1) A is said to be of class Σ(θ + π/2, R) if there are positive constants θ, R such that 0 < θ < π/2, and Σ(θ + π/2, R) ⊂ ρ(A) and

sup λ∈Σ(θ+π/2,R)

kλR(λ, A)k < ∞,

(1.25)

(2) A and B are said to satisfy condition P if there are positive constants θ, θ0 , R, θ0 < θ such that A and B are of class Σ(θ + π/2, R), Σ(π/2 − θ0 , R), respectively. If A and B are commuting operators, A + B is defined by (A + B)x = Ax + Bx with domain D(A + B) = D(A) ∩ D(B). We will use the following norm, defined by A on the space X, kxkTA := kR(λ, A)xk, where λ ∈ ρ(A). It is seen that different λ ∈ ρ(A) yields equivalent norms. We say that an operator C on X is A-closed if its graph is closed with respect to the topology induced by TA on the product X × X. It is easily seen that C is A-closable if xn → 0, xn ∈ D(C), Cxn → y with respect to TA in X implies y = 0. In this case, A-closure of C is denoted A by C . Theorem 1.11. Assume that A and B commute. Then the following assertions hold: (1) If one of the operators is bounded, then σ(A + B) ⊂ σ(A) + σ(B).

(1.26)

(2) If A and B satisfy condition P, then A + B is A-closable, and A

σ((A + B) ) ⊂ σ(A) + σ(B).

(1.27) A

In particular, if D(A) is dense in X, then (A + B) = A + B , where A + B denotes the usual closure of A + B. Proof. For the proof we refer the reader to Theorems 7.2, 7.3 in [Arendt, R¨ abiger and Sourour (6)].

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1.3 1.3.1

19

Spectral Theory and Almost Periodicity of Functions Introduction

As is known, for a 2π-periodic continuous function f : R → X the Fourier exponents are defined to be the set: Z π e−iλξ f (ξ)dξ 6= 0}. (1.28) {λ ∈ Z : −π

The notion of spectrum of a bounded function is a generalization of this notion of Fourier exponents. However, for any bounded function it is not expected that the integral in the above set is used, but instead of it another integral on the whole real line. This section will give a very short introduction to this spectral theory. We will take several examples to show how our abstract definition can be incorporated into simple cases in which the notions of Fourier, Bohr exponents are well known. 1.3.2

Spectrum of a Bounded Function

We denote by F the Fourier transform, i.e. ˆ := f(s)

Z

+∞

e−ist f (t)dt

(1.29)

−∞

(s ∈ R, f ∈ L1 (R)). Then the Beurling spectrum of u ∈ BU C(R, X) is defined to be the following set sp(u) := {ξ ∈ R : ∀ε > 0∃f ∈ L1 (R), suppfˆ ⊂ (ξ − ε, ξ + ε), f ∗ u 6= 0} (1.30) where Z +∞ f ∗ u(s) := f (s − t)u(t)dt. −∞

We consider the simplest case Example 1.14. Let f (t) = aetλt , where λ ∈ R, a ∈ X. Then sp(f ) = λ. Proof. As is well known, for any real µ 6= λ there exists a function φ ∈ L1 (R) such that the support of the Fourier transform of φ is contained in the interval [µ − ε, µ + ε] for any ε such that λ 6∈ [µ − ε, µ + ε]. Hence,

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the convolution ψ(s) =

Z

=a

∞

φ(s − t)f (t)dt =

−∞ Z ∞

φ(s − t)eiλt dt

−∞ Z ∞

= −a

= −ae

−∞

iλs

Z

φ(ξ)eiλξ+s dξ ∞

φ(ξ)eiλξ

−∞

ˆ = −aeiλs φ(λ)

= aeiλs × 0 = 0.

(1.31)

This shows that sp(f ) ⊂ {λ}. On the other hand, if in the above argument ˆ we take µ = λ and φ ∈ L1 (R) such that φ(λ) = 1 , then ψ(s) = aeiλs 6= 0. This yields that sp(f ) = {λ}. As an immediate consequence of this example, the following holds true: Example 1.15. Let f (t) =

N X

ak eiλk t ,

k=0

where ak 6= 0, λk ∈ R, ∀k = 1, 2, ..., N. Then sp(f ) = {λ1 , ..., λN }. Example 1.16. If f (t) is a 2π-periodic function with Fourier series X ak e2iπkt , k∈Z

then sp(f ) = {2πk : ak 6= 0}. Proof. For every λ 6= 2k0 π, k0 ∈ Z or λ = 2k0 π at which fk0 = 0, where fn is the Fourier coefficients of f , and for every positive ε, let φ ∈ L1 (R) be a complex valued continuous function such that the support of its Fourier ˆ ⊂ [λ − ε, λ + ε]. Put transform suppφ(ξ) Z ∞ u(t) = f ∗ φ(t) = f (t − s)φ(s)ds. −∞

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Since f is periodic, there is a sequence of trigonometric polynomials N (n)

Pn (t) =

X

an,k e2ikπt

k=1

that is convergent uniformly to f with respect to t ∈ R such that limn→∞ an,k = fn . We have u(t) = f ∗ φ(t) = lim Pn ∗ φ(t) n→∞

N (n)

= lim

n→∞

= lim

n→∞

= lim

n→∞

= 0.

X

an,k e2ikπ· ∗ φ(t)

X

an,k e2ikπt

X

ˆ an,k e2ikπt φ(2kπ)

k=1 N (n)

k=1 N (n)

Z

∞

e−2ikπs φ(s)ds

−∞

k=1

This, by definition, shows that sp(f ) ⊂ {m ∈ 2πZ : fm 6= 0}. Conversely, for λ ∈ {m ∈ 2πZ : fm 6= 0} and for every sufficiently small positive ε we can choose a complex function ϕ ∈ L1 (R) such that ϕ(ξ) ˆ = 1, ∀ξ ∈ [λ − ε/2, λ + ε/2] and ϕ(ξ) ˆ = 0, ∀ξ 6∈ [λ − ε, λ + ε]. Repeating the above argument, we have w(t) = f ∗ ϕ(t) = lim Pn (t) ∗ ϕ(t) n→∞

N (n)

= lim

n→∞

X

an,k e2ikπt ϕ(2kπ) ˆ

k=1

= lim an,k0 e2ik0 πt . n→∞

Since limn→∞ an,k0 = fk0 this shows that w 6= 0. Thus, λ ∈ sp(f ).

(1.32)

Exercise 8. Let f (t) = ae−|t| . Show that sp(f ) = R. Exercise 9. Let f be a continuous function with compact support. Show that Z ∞ sp(f ) = {ρ ∈ R : f (t)e−iρt dt 6= 0}. −∞

From this show that if f is positive and has compact support, then 0 6∈ sp(f ).

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There is another way to approach the notion of spectrum of a bounded function via the Fourier- Carleman transform of a bounded function u defined by the formula

u ˆ(λ) =

R ∞ −λt    0 e u(t)dt, (Reλ > 0);

(1.33)

  − R ∞ eλt u(−t)dt, (Reλ < 0). 0

In fact, we will define the notion of Carleman spectrum of a bounded continuous function u as the set σ(u) of all reals λ at which the FourierCarleman transform has a no holomorphic extension to any neighborhood of iλ. In fact, we compute σ(u) in several simplest cases. Let u(t) = aeiλ0 t , λ0 ∈ R, a 6= 0. Then, for Reλ > 0 u ˆ(λ) = =

Z Z

∞ 0

∞

e−λt u(t)dt e−λt aeiλ0 t dt

0

1 1 e(iλ0 −λ)t − t→+∞ iλ0 − λ iλ0 − λ a =− . iλ0 − λ =a

lim

Similarly, for Reλ < 0 we can compute u ˆ(λ) which is of the same form. Hence, u ˆ(λ) has holomorphic extension at any iξ 6= iλ0 . Obviously, σ(u) = {λ0 }. We consider now a more general case in which f is a τ -periodic continuous function. Example 1.17. Let f be a X-valued τ -periodic continuous function. Then

σ(f ) = {2πn/τ | n ∈ Z,

Z

τ 0

e−i2πnt/τ f (t)dt 6= 0}.

(1.34)

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By definition, for Reλ > 0, Z ∞ ˆ e−λt f (t)dt f (λ) = 0

= = = =

∞ Z X

nτ

n=1 ∞ Z τ X n=1 ∞ X

n=1 Z τ

e−λt f (t)dt

(n−1)τ

0

e−λ(t+(n−1)τ ) f (t + (n − 1)τ )dt

e−(n−1)λτ

Z

e−λt f (t)dt

0

=

Z

τ

e−λt f (t)dt

0

∞ X

e−(n−1)λτ

n=1

τ

e

−λt

f (t)dt

0

1 . 1 − e−λτ

(1.35)

Similarly, for Reλ < 0, (1.35) holds true as well. From (1.35) it is seen that if λ is such that e−λτ 6= 1, i.e, λ 6= 2πin/τ for n ∈ Z, then λ 6∈ σ(f ) because at this point fˆ(λ) has a holomorphic extension.RMoreover, at λn = 2πin/τ , τ fˆ(λ) has a holomorphic extension if and only if 0 e−2πnt/τ f (t)dt = 0. This shows that (1.34) holds true. We have just shown that σ(f ) = sp(f ) for periodic functions. In general, for bounded continuous functions they coincide with each other. Theorem 1.12. Under the notation as above, sp(u) coincides with the set σ(u). Proof. (91)].

For the proof we refer the reader to Proposition 0.5, p.22 in [Pruss

Every definition of spectrum has its advantages. We will see this in the next chapter. Below we collect some main properties of the spectrum of a function, which we will need in the sequel. Theorem 1.13. Let f, gn ∈ BU C(R, X), n ∈ N such that gn → f as n → ∞. Then (i) sp(f ) is closed, (ii) sp(f (· + h)) = sp(f ), (iii) If α ∈ C\{0} sp(αf ) = sp(f ),

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(iv) If sp(gn ) ⊂ Λ for all n ∈ N then sp(f ) ⊂ Λ, (v) If A is a closed operator, f (t) ∈ D(A)∀t ∈ R and Af (·) ∈ BU C(R, X), then, sp(Af ) ⊂ sp(f ), (vi) sp(ψ ∗ f ) ⊂ sp(f ) ∩ suppFψ, ∀ψ ∈ L1 (R). Proof. The proofs of (i)-(iii) are straightforward. We refer the reader to Proposition 0.4, p. 20, Theorem 0.8 , p. 21 in [Vu (102)] and pp. 20-21 in [Pruss (91)] for the proofs of the remaining assertions. As a consequence of Theorem 1.13 we have the following: Corollary 1.2. Let Λ ⊂ R be closed. Then, the set {f ∈ BU C(R, X) : sp(f ) ⊂ Λ}

(1.36)

is a closed subspace of BU C(R, X). We consider the translation group (S(t))t∈R on BU C(R, X). One of the frequently used properties of the spectrum of a function is the following: Theorem 1.14. Under the notation as above, i sp(u) = σ(Du ),

(1.37)

where Du is the generator of the restriction of the group S(t) to Mu := span{S(t)u, t ∈ R}. Proof. 1.3.3

For the proof see Theorem 8.19, p. 213 in [Davies (27)].

Uniform Spectrum of a Bounded Function

Notice that for every λ ∈ C with <λ 6= 0 and f ∈ BC(R, X) the function \ ϕf (λ) : R 3 t 7→ S(t)f(λ) ∈ X belongs to Mf ⊂ BC(R, X). Moreover, ϕf (λ) is analytic on C\iR. Definition 1.12. Let f be in BC(R, X). Then, (1) α ∈ R is said to be uniformly regular with respect to f if there exists a neighborhood U of iα in C such that the function ϕf (λ), as a complex function of λ with <λ 6= 0, has an analytic continuation into U. (2) The set of ξ ∈ R such that ξ is not uniformly regular with respect to f ∈ BC(R, X) is called uniform spectrum of f and is denoted by spu (f ).

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If f ∈ BU C(R, X), then α ∈ R is uniformly regular if and only if it is regular with respect to f . In fact, this follows from the fact that for bounded uniformly continuous functions u, the identity (1.37) holds. Next, using the identity Z ∞ R(λ, Du )u = e−(λ)ξ S(ξ)udξ, <λ 6= 0 0

we get the claim. For f ∈ BC(R, X), in general, the above (1.37) may not hold. We now study properties of uniform spectra of functions in BC(R, X). Proposition 1.1. Let g, f, fn ∈ BC(R, X) such that fn → f as n → ∞ and let Λ ⊂ R be a closed subset. Then the following assertions hold: (i) (ii) (iii) (iv) (v) (vi)

spu (f ) = spu (f (h + ·)); spu (αf (·)) ⊂ spu (f ), α ∈ C; sp(f ) ⊂ spu (f ); spu (Bf (·)) ⊂ spu (f ), B ∈ L(X); spu (f + g) ⊂ spu (f ) ∪ spu (g); spu (f ) ⊂ Λ.

Proof. (i) - (v) are obvious from the definitions of spectrum and uniform spectrum. Now we prove (vi). Let ρ0 6∈ Λ. Since Λ is closed, there is a positive constant r < dist(ρ0 , Λ). We can prove that since kϕfn (λ)k ≤

2kf k , |<λ|

¯r (iρ0 ) ∀λ ∈ B

(1.38)

¯r (iρ0 ), n ≥ N. ∀λ ∈ B

(1.39)

for sufficiently large n ≥ N , one has kϕfn (λ)k ≤

4kf k , 3r

Obviously, for every fixed λ such that <λ 6= 0 we have ϕfn (λ) → ϕf (λ). Now applying Vitali Theorem to the sequence of complex functions {ϕfn } we see that ϕfn is convergent uniformly on Br (iρ0 ) to ϕf . This yields that ϕf is holomorphic on Br (iρ0 ), that is ρ0 is a uniformly regular point with respect to f and ρ0 6∈ spu (f ). As an immediate consequence of (iii) of the above proposition, we have Corollary 1.3. For any closed subset Λ ⊂ R, the set Λu (X) := {f ∈ BC(R, X) : spu (f ) ⊂ Λ} is a closed subspace of BC(R, X) which is invariant under translations.

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The following result will be needed in the sequel. Lemma 1.2. Let Λ be a closed subset of R and let DΛu be the differential operator acting on Λu (X). Then we have σ(DΛu ) = iΛ.

(1.40)

Proof. Since the function gα defined by gα (t) := eiαt x, α ∈ R, t ∈ R, x 6= 0, is in Λu (X) and spu (gα ) = sp(gα ) = {α} we see that iα ∈ σ(DΛu ), that is, iΛ ⊂ σ(DΛu ). Now we prove the converse. For β ∈ R\Λ we consider the equation iβg − g 0 = f,

f ∈ Λu (X).

(1.41)

We will prove that (1.41) is uniquely solvable for every f ∈ Λu (X). This equation has at most one solution. In fact, if g1 , g2 are two solutions, then g = g1 − g2 is a solution of the homogeneous equation, that is for f = 0. Taking Carlemann transform of both sides of the correspoding equation we may see that sp(g) ⊂ {β}. Since g ∈ Λu (X) we have sp(g) ⊂ Λ. Combining these facts we have sp(g) = ∅, that is g = 0. Now we prove the existence of at least one solution to Eq. (1.41). For <λ 6= 0 Eq. (1.41) has a unique solution which is nothing but ϕf (λ), so by definition, ϕf (λ) = (λ − Df )−1 f,

<λ 6= 0.

Using a similar argument as in the proof of (iii) of Proposition 1.1 we ¯r (iβ) uniformly in u ∈ can show that (λ − Df )−1 u is bounded on B span{S(h)f, h ∈ R}, kuk ≤ 1 for certain positive constant r independent of u and λ. Since iβ is a limit point of σ(Df ), this boundedness yields in particular that iβ ∈ ρ(Df ). Hence, there exists a unique solution g ∈ Mf ⊂ Λu (X) to (1.41). 1.3.4

Almost Periodic Functions

1.3.4.1 Definition and basic properties A subset E ⊂ R is said to be relatively dense if there exists a number l > 0 (inclusion length) such that every interval [a, a + l] contains at least one point of E. Let f be a function on R taking values in a complex Banach space X. f is said to be almost periodic if to every ε > 0 there corresponds a relatively dense set T (ε, f ) (of ε-translations, or ε-periods ) such that sup kf (t + τ ) − f (t)k ≤ ε, ∀τ ∈ T (ε, f ). t∈R

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A typical example of an almost periodic function that is not periodic is the following: Example 1.18. f (t) = a sin t + a sin

√ 2t , ∀ 0 6= a ∈ X.

(1.42)

In general, the function f (t) = aeit + bei

√ 2t

, a, b ∈ X, a 6= 0, b 6= 0,

is an almost periodic one that is not periodic. Proof. We will make use of only the definitions and the following fact from the elementary mathematics:√for every √ constant ε > 0 there exists a positive integer N (ε) such that N 2 − [N 2] < ε, where [r] denotes the integer part of the real number r. By this fact, for a positive ε there is an interval (2M π − α, 2M π + α), where M is an integer and α > 0, such that √ √ ε kbei 2(t+τ ) − bei 2t k < , ∀t, τ ∈ (2M π − α, 2M π + α). 2 Hence, for sufficiently small α and l = 2M π every inteval of length l contains at least an ε-period of the function f . This shows that f is almost periodic. Now we are going to prove that f is not periodic. In fact, suppose to the contrary that f is periodic with period T . By this assumption, the function g(t) := aei(t+T ) + bei

√ 2(t+T )

− aeit − bei

√ 2t

= 0 ∀t ∈ R,

Thus, 0=

Z

2π

g(t)dt = b 0

Z

2π

(ei

√ 2(t+T )

0

− ei

√ 2t

)dt

√ √ 1 = √ (ei 22π − 1)(ei 2T − 1). i 2

(1.43)

√ This shows that T / 2 must be rational. Similarly, we can show that T is rational. This leads to a contradiction showing that f is not periodic. Generally, the sum of almost periodic functions are an almost periodic function. Example 1.19. All trigonometric polynomials P (t) =

n X

k=1

are almost periodic.

ak eiλk t , (ak ∈ X, λk ∈ R)

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We collect some basic properties of an almost periodic function in the following: Theorem 1.15. Let f and fn , n ∈ R be almost periodic functions with values in X. Then the following assertions hold true: (1) The range of f is precompact, i.e., the set {f (t), t ∈ R} is a compact subset of X, so f is bounded; (2) f is uniformly continuous on R; (3) If fn → g as n → ∞ uniformly, then g is almost periodic; (4) If f 0 is uniformly continuous, then f 0 is almost periodic. Proof.

For the proof see e.g. pp. 5-6 [Amerio and Prouse (3)].

As a consequence of Theorem 1.15 the space of all almost periodic functions taking values in X with sup-norm is a Banach space which will be denoted by AP (X). For almost periodic functions the following criterion holds (Bochner’s criterion): Theorem 1.16. Let f be a continuous function taking values in X. Then f is almost periodic if and only if given a sequence {cn }n∈N there exists a subsequence {cnk }k∈N such that the sequence {f (t + cnk )}k∈N converges uniformly. Proof.

For the proof see e.g. p. 9 in [Amerio and Prouse (3)].

Exercise 10. Let f ∈ BU C(R, X) such that eisp(f ) is finite. Show that f is of the form (1.44), so f is almost periodic. Proof.

By Theorem 1.14 eisp(f ) = eσ(Df ) .

On the other hand, by the Weak Spectral Mapping Theorem, eσ(Df ) = σ(S(1)|Mf ). Hence, using Riesz Integral we can decompose Mf into the direct sum of finite closed subspaces M1 , ..., Mk invariant under the translation group (S(t))t∈R . Moreover, the spectrum of S(1) restricted to every subspace consists of only one point. By Gelfand Theorem, S(1) should have the following form: S(1)|Mj = eiλj I, λj ∈ R. It is easy to see that if, by the

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above decomposition, f = f1 +...+fk then the function gj (t) := e−iλj t fj (t) satisfies gj (t + 1) = e−iλj (t+1) fj (t + 1) = e−iλj t e−iλj eiλj fj (t) = e−iλj t eiλj fj (t) = gj (t), ∀t ∈ R, j = 1, 2, ..., k. Finally, f (t) =

k X

eiλt gj (t),

(1.44)

j=1

where gj (·) is 1-periodic. This shows that f is almost periodic. 1.3.5

Sprectrum of an Almost Periodic Function

There is a natural extension of the notion of Fourier exponents of periodic functions to almost periodic functions. In fact, if f is almost periodic function taking values in X, then for every λ ∈ R the average Z T 1 e−iλt f (t)dt a(f, λ) := lim T →∞ 2T −T

exists and is different from 0 at most at countably many points λ. The set {λ ∈ R : a(f, λ) 6= 0} is called Bohr spectrum of f which will be denoted by σb (f ) . The following Approximation Theorem of almost periodic functions holds Theorem 1.17. (Approximation Theorem) Let f be an almost periodic function. Then for every ε > 0 there exists a trigonometric polynomial Pε (t) =

N X j=1

such that

aj eiλj t , aj ∈ X, λj ∈ σb (f )

sup kf (t) − Pε (t)k < ε. t∈R

Proof.

For the proof see e.g. pp. 17-24 in [Levitan and Zhikov (58)].

Remark 1.3. The trigonometric polynomials Pε (t) in Theorem 1.17 can be chosen as an element of the space Mf := span{S(τ )f, τ ∈ R}

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(see p. 29 in [Levitan and Zhikov (58)]. Moreover, without loss of generality by assuming that σb (f ) = {λ1 , λ2 , · · · } one can choose a sequence of trigonometric polynomials, called trigonometric polynomials of BochnerFejer, approximating f such that N (m)

Pm (t) =

X j=1

γm,j a(λj , f )eiλj t , m ∈ N,

where limm→∞ γm,j = 1. As a consequence we have: Corollary 1.4. Let f be almost periodic. Then Mf = span{a(λ, f )eiλ· , λ ∈ σb (f )}. Proof.

By Theorem 1.17, Mf ⊂ span{a(f, λ)eiλ· , λ ∈ σb (f )}.

On the other hand, it is easy to prove by induction that if P is any trigonometric polynomial with different exponents {λ1 , · · · , λk }, such that k X xj eiλk t , P (t) = j=1

then xj eiλj ∈ MP , ∀j = 1, · · · , k. a(λj , f )eiλj ∈ Mf , ∀j ∈ N.

Hence by Remark 1.3, obviously,

The relation between the spectrum of an almost periodic function f and its Bohr spectrum is stated in the following: Proposition 1.2. If f is an almost periodic function, then sp(f ) = σb (f ). Proof. Let λ ∈ σb (f ). Then there is a x ∈ X such that xeiλ· ∈ Mf . Obviously, λ ∈ σ(D|Mf ). By Theorem 1.14 λ ∈ sp(f ). Conversely, by Theorem 1.17, f can be approximated by a sequence of trigonometric polynomials with exponents contained in σb (f ). In view of Theorem 1.13 sp(f ) ⊂ σb (f ). 1.3.6

A Spectral Criterion for Almost Periodicity of a Function

Suppose that we know beforehand that f ∈ BU C(R, X). It is often possible to establish the almost periodicity of this function starting from certain a priori information about its spectrum. Theorem 1.18. Let E and G be closed, translation invariant subspaces of BU C(R, X) and suppose that

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31

G ⊂ E; G contains all constant functions which belong to E; iξ· E and G are invariant under multiplications R t by e for all ξ ∈ R; whenever f ∈ G and F ∈ E, where F (t) = 0 f (s)ds, then F ∈ G.

Let u ∈ E have countable reduced spectrum

spG := {ξ ∈ R : ∀ε > 0∃f ∈ L1 (R) such that

suppFf ⊂ (ξ − ε, ξ + ε) and f ∗ u 6∈ G}.

Then u ∈ G. Proof.

For the proof see p. 371 in [Arendt and Batty (5)].

Remark 1.4. In the case where G = AP (X) the condition iv) in Theorem 1.18 can be replaced by the condition that X does not contain c0 (see Proposition 3.1, p. 369 in [Arendt and Batty (5)]). Another alternative of the condition iv) is the total ergodicity of u which is defined as follows: u ∈ BU C(R, X) is called totally ergodic if Z τ 1 eiηs S(s)ds Mη u := lim τ →∞ 2τ −τ exists in BU C(R, X) for all η ∈ R. From this remark the following example is obvious: Example 1.20. A function f ∈ BU C(R, X) is 2π-periodic if and only if sp(f ) ⊂ 2πZ. 1.3.7

Almost Automorphic Functions

Definition and Basic Properties. A function f ∈ C(R, X) is said to be almost automorphic if for any sequence of real numbers (s0n ), there exists a subsequence (sn ) such that lim lim f (t + sn − sm ) = f (t)

m→∞ n→∞

(1.45)

for any t ∈ R. The limit in (1.45) means g(t) = lim f (t + ss ) n→∞

is well-defined for each t ∈ R and

(1.46)

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f (t) = lim g(t − sn ) n→∞

(1.47)

for each t ∈ R. Remark 1.5. Because of pointwise convergence the function g is measurable but not necessarily continous. Remark 1.6. It is also clear from the definition above that constant functions and continuous almost periodic functions are almost automorphic. If the limit in (1.41) is uniform on any compact subset K ⊂ R, we say that f is compact almost automorphic. Theorem 1.19. Assume that f , f1 , and f2 are almost automorphic functions taking values in a Banach space X, φ is a scalar almost automorphic function, and λ is any scalar, then the following hold true. (i) (ii) (iii) (iv) (v)

λf and f1 + f2 are almost automorphic, fτ (t) := f (t + τ ), t ∈ R is almost automorphic; ¯ := f (−t), t ∈ R is almost automorphic; f(t)

The Range Rf of f is precompact, so f is bounded; The function t 7→ φ(t)f (t) is almost automorphic.

Proof. See Theorems 2.1.3 and 2.1.4 in [N’Gu´er´ekata (79)], for the proofs of (i)-(iv). The proof of (v) is straightforward, and is left to the reader. Theorem 1.20. If {fn } is a sequence of almost automorphic X-valued functions such that fn → f uniformly on R, then f is almost automorphic. Proof.

See Theorem 2.1.10 in [N’Gu´er´ekata (79)], for proof.

Remark 1.7. If we equip AA(X), the space of almost automorphic functions with the sup norm kf k∞ = sup kf (t)k t∈R

then it turns out to be a Banach space. If we denote KAA(X), the space of compact almost automorphic X-valued functions, then we have AP (X) ⊂ KAA(X) ⊂ AA(X) ⊂ BC(R, X) .

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Theorem 1.21. If f ∈ AA(X), then i) kf k∞ = kgk∞ ii) Rg ⊂ Rf where g is the function defined in (1.41)-(1.42). Proof.

i) Using (1.41) we may write kg(t)k ≤ kg(t) − f (t + sn )k + kf (t + sn )k

and choosing n large enough, we get kg(t)k < ε + sup kf (σ)k σ∈R

Hence sup kg(t)k ≤ sup kf (t)k t∈R

t∈R

Similarly by (1.42), we obtain sup kf (t)k ≤ sup kg(t)k t∈R

(1.48)

t∈R

which proves the theorem. ii) This statement is straight forward.

Theorem 1.22. If f ∈ AA(X) and its derivative f 0 exists and is uniformly continuous on R, then f 0 ∈ AA(X). Proof. It suffices observe that for each n ∈ N, n(f (t + n1 ) − f (t)) is an almost automorphic function and the sequence of these functions converges uniformly to f 0 on R (see Theorem 2.4.1 in [N’Gu´er´ekata (79)] for a detailed proof). Rt Theorem 1.23. Let us define F : R 7→ X by F (t) = 0 f (s)ds where f ∈ AA(X). Then F ∈ AA(X) iff RF = {F (t)/t ∈ R} is precompact. Before we prove the Theorem, let us introduce some useful notations (due to S. Bochner). Remark 1.8. If f : R → X is a function and a sequence of real numbers s = (sn ) is such that we have lim f (t + sn ) = g(t), pointwise on R,

n→∞

we will write Ts f = g.

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Remark 1.9. i) Ts is a linear operator. Indeed, given a fixed sequence s = (sn ) ⊂ R, the domain of Ts is D(Ts ) = {f : R → X / Ts f exists}. D(Ts ) is a linear set for if f, f1 , f2 ∈ D(Ts ), then f1 + f2 ∈ D(Ts ) and λf ∈ D(Ts ) for any scalar λ. And obviously, Ts (f1 + f2 ) = Ts f1 + Ts f2 and Ts (λf ) = λTs f . ii) Let us write −s = (−sn ) and suppose that f ∈ D(Ts ) and Ts f ∈ D(T−s ). Then the product operator As = T−s Ts f is well defined. It is easy to verify that As is also a linear operator. iii) As maps bounded functions into bounded functions, and for almost automorphic functions f , we get As f = f . We are now ready to prove the previous Theorem: Proof. It suffices to prove that F (t) is almost automorphic if RF is precompact. Let (s00n ) be a sequence of real numbers. Then there exists a subsequence (s0n ) such that lim f (t + s0n ) = g(t)

n→∞

and lim g(t − s0n ) = f (t),

n→∞

pointwise on R, and lim F (s0n ) = α1 ,

n→∞

for some vector α1 ∈ X. We get for every t ∈ R: Z t+s0n Z F (t + s0n ) = f (r) dr = 0

s0n

f (r) dr + 0

= F (s0n ) +

Z

Z

t+s0n

f (r) dr s0n

t+s0n

f (s) dr. s0n

Using the substitution σ = r − s0n , we obtain Z t F (t + s0n ) = F (s0n ) + f (σ + s0n ) dσ. 0

If we apply the Lebesgue’s dominated convergence theorem, we obtain Z t 0 lim F (t + sn ) = α1 + g(σ) ˙ dσ n→∞

for each t ∈ R.

0

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Let us observe that the range of the function G(t) = α1 + also precompact and

Rt 0

g(r) dr is

sup kG(t)k = sup kF (t)k t∈R

t∈R

by Theorem 1.21 i), so that we can extract a subsequence (sn ) of (s0n ) such that lim G(−sn ) = α2 ,

n→∞

for some α2 ∈ X. Now we can write G(t − sn ) = G(−sn ) +

Z

t 0

g(r − sn ) dr

so that lim G(t − sn ) = α2 +

n→∞

Z

t

f (r) dr = α2 + F (t). 0

Let us prove now that α2 = θ. Using the notation above we get As F = α2 + F, where s = (sn ). Now it is easy to observe that F as well as α2 belong to the domain of As ; therefore As F also is in the domain of As and we deduce the equation A2s F = As α2 + As F = α2 + α2 + F = 2α2 + F We can continue indefinitely the process to get Ans F = nα2 + F,

∀n = 1, 2, . . . .

But we have sup kAns F (t)k ≤ sup kF (t)k t∈R

t∈R

and F (t) is a bounded function. This leads to a contradiction if α2 6= 0. Hence, α2 = 0 and As F = F ; so F ∈ AA(X) . The proof is complete. Remark 1.10. If X is a uniformly convex Banach space, the assumption on RF can be weakened. Indeed, the result holds true if RF is bounded (see, Theorem 2.4.4 and Theorem 2.4.6 in [N’Gu´er´ekata (79)] ).

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We can recall this other important result: Theorem 1.24. Let (T (t))t∈R be a C0 -group of bounded linear operators, and assume that the function x(t) = T (t)x0 : R 7→ X, where x(0) = x0 ∈ X is almost automorphic. Then either inf t∈R kx(t)k > 0, or x(t) = 0 for every t ∈ R. Proof. Assume that inf t∈R kx(t)k = 0. Then there exists a minimizing sequence (s0n ) such that kx(s0n )k 7→ 0 as n 7→ ∞. Since x(t) is almost automorphic, there exists a subsequence (sn ) ⊂ (s0n ) such that lim x(t + sn ) = y(t)

n→∞

exists for every t ∈ R, and lim y(t − sn ) = x(t)

n→∞

for every t ∈ R. We have x(t + sn ) = T (t + sn ) − T (t)T (sn )x0 = T (t)x(sn ) . So, ky(t)k = lim kT (t)x(sn )k ≤ lim kT (t)kkx(sn )k = 0 n→∞

n→∞

for every t ∈ R. We infer that x(t) = 0 for every t ∈ R. The theorem is proved. Definition 1.13. A function f ∈ C(R+ , X) is said to be asymptotically almost automorphic, if there exists g ∈ AA(X) and h ∈ C(R+ , X) with the property that limt→∞ kh(t)k = 0, such that f (t) = g(t) + h(t), t ∈ R+ .

(1.49)

The functions g and h are called respectively the principal and the corrective terms of f . Theorem 1.25. If f is asymptotically almost automorphic then its principal and corrective terms are uniquely determined. Proof.

See Theorem 2.5.4 in [N’Gu´er´ekata (79)].

Exercise 11. Prove that every asymptotically almost function is bounded over R+ .

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Exercise 12. Let f ∈ C(R+ , X) and ν ∈ C(R+ , C) be asymptotically almost automorphic. Show that fτ (t) := f (t + τ ), for a fixed τ ∈ R+ and (νf )(t) = ν(t)f (t) are also asymptotically almost automorphic. Exercise 13. Let AAA(X) be the space of asymptotically almost automorphic functions with the norm kf kAAA(X) = kgkAA(X) + khkC(R+,X) , where g and h are respectiveley the prinicipal and the corrective terms of f . Show that if (fn ) is a sequence of functions in AAA(X) that converges uniformly to f , then f ∈ AAA(X).

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Chapter 2

Stability and Exponential Dichotomy of Solutions of Homogeneous Equations This chapter is devoted to the study of the behavior of homogeneous equations of the form u0 (t) = A(t)u(t), t ∈ R. The exponential dichotomy and stability will be characterized in terms of Perron’s criteria.

2.1

Perron Theorem

In this section we will extend the Perron Theorem for ordinary differential equations to equations in Banach spaces. Throughout this section we consider only processes defined on the whole line. We denote ∆ := {(t, s) ∈ R : t ≥ s}. Definition 2.1. A family of bounded linear operators (U (t, s))t≥s , t, s ∈ R from a Banach space X to itself is called a strongly continuous evolutionary process if the following conditions are satisfied (1) (2) (3) (4)

U (t, t) = I for all t ∈ R, U (t, s)U (s, r) = U (t, r) for all t ≥ s ≥ r, The map ∆ 3 (t, s) 7→ U (t, s)x is continuous for every fixed x ∈ X, kU (t, s)k ≤ N eω(t−s) for some positive N, ω independent of (t, s) ∈ ∆.

Definition 2.2. Let (U (t, s))t≥s be a strongly continuous evolutionary process on a Banach space X. (U (t, s))t≥s is said to have an exponential dichotomy if there exist a family of projections Q(t), t ∈ R and positive constants M, α such that the following conditions are satisfied: (1) For every fixed x ∈ X the map t 7→ Q(t)x is continuous, (2) Q(t)U (t, s) = U (t, s)Q(s), ∀(t, s) ∈ ∆, (3) kU (t, s)xk ≤ M e−α(t−s) kxk, ∀(t, s) ∈ ∆, x ∈ KerQ(s), 39

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(4) kU (t, s)yk ≥ M −1 eα(t−s) kyk, ∀(s, t) ∈ ∆, y ∈ ImQ(s), (5) U (t, s)|ImQ(s) is an isomorphism from ImQ(s) onto ImQ(t) , ∀(t, s) ∈ ∆. In this section we will establish a relation between the exponential dichotomy of a given strongly continuous process (U (t, s))t≥s and the unique solvability of the following integral equation in appropriate function spaces: Z t u(t) = U (t, s)u(s) + U (t, ξ)f (ξ)dξ, ∀(t, s) ∈ ∆. (2.1) s

Next, we define an operator L : D(L) ⊂ C0 (R, X) → C0 (R, X) as follows: u ∈ C0 (R, X) if there exists a function f ∈ C0 (R, X) such that (2.1) holds. For u ∈ D(L) we define Lu = f . Lemma 2.1. The operator L is a well-defined, closed and linear operator. Proof.

To prove the well-definedness of the operator we suppose that Z t u(t) := U (t, s)u(s) + U (t, ξ)g(ξ)dξ, ∀(t, s) ∈ ∆ s

u(t) := U (t, s)u(s) +

Z

t

U (t, ξ)f (ξ)dξ, s

∀(t, s) ∈ ∆.

Then, Z

t

U (t, ξ)g(ξ)dξ = s

Z

t

U (t, ξ)f (ξ)dξ. s

Hence, Z

t s

U (t, ξ)[f (ξ) − g(ξ)]dξ = 0.

Thus, 1 (t − s)

Z

t s

U (t, ξ)[f (ξ) − g(ξ)]dξ = 0.

Letting t − s → 0, by the strong continuity of the process (U (t, s))t≥s we obtain that f (t) = g(t), So, L is well-defined.

t ∈ R.

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Let {vn } be a sequence in D(L), such that limn→∞ kvn − vk = 0 for some v ∈ C0 (R, X) and ∃f ∈ C0 (R, X) such that limn→∞ kLvn − f k = 0. Now we prove that v ∈ D(L) and Lv = f . In fact, we have Z t vn (t) = U (t, s)vn (s) + U (t, ξ)Lvn (ξ)dξ, (t, s) ∈ ∆. (2.2) s

For fixed (t, s) ∈ ∆, we have Z t Z t Z t k U (t, ξ)Lvn (ξ)dξ − U (t, ξ)f (ξ)dξk ≤ kU (t, ξ)k.kLvn (ξ)−f (ξ)kdξ s

s

≤ N1

Z

s

t s

kLvn (ξ) − f (ξ)kdξ ≤ N1 (t − s)kLvn − f k

From this we obtain Z t Z t lim k U (t, ξ)Lvn (ξ)dξ − U (t, ξ)f (ξ)dξk = 0. n→∞

s

(2.3)

s

This yields

v(t) = U (t, s)v(s) +

Z

t

U (t, ξ)f (ξ)dξ, s

(t, s) ∈ ∆.

Therefore, v ∈ D(L) and Lv = f . The linearity of the operator L can be easily shown. Theorem 2.1. Let (U (t, s))t≥s be a given strongly continuous evolutionary process. Then the following assertions are equivalent: (i) The process (U (t, s))t≥s has an exponential dichotomy; (ii) For every given f ∈ C0 (R, X) the integral equation (2.1) has a unique solution xf ∈ C0 (R, X); (iii) For every given bounded and continuous function f the integral equation (2.1) has a unique bounded solution. Proof. We first prove the equivalence between (i) and (ii). The equivalence between (i) and (ii) can be established in the same way. ”(i) ⇒ (ii)”: Let P (t) := I − Q(t), t ∈ R, where Q(t) are determined from the exponential dichotomy of the process (U (t, s))t≥s . We now show that the equation Lu = 0 has the only trivial solution u = 0. In fact, by the definition of exponential dichotomy (Definition 2.2), we have kP (t)u(t)k = kU (t, s)P (s)u(s)k ≤ M e−α(t−s) ku(s)k ≤ M e−α(t−s) kuk,

∀t ≥ s.

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Hence, letting s → −∞ we get P (t)u(t) = 0, ∀t ∈ R. Similarly, we can show Q(t)u(t) = 0, ∀t ∈ R. This yields that u(t) = 0, ∀t ∈ R. Next, we define a so-called Green function ( U (t, s)P (s), t ≥ s G(t, s) = −U (t, s)Q(s), t < s.

By the exponential dichotomy, we see that there are positive constants K, α such that kG(t, s)k ≤ Ke−α|t−s| ,

∀t, s ∈ R.

We claim that the function u(t) =

Z

∞

t∈R

G(t, ξ)f (ξ)dξ,

−∞

is a (unique) bounded continuous solution to the integral equation (2.1). In fact, for any t ≥ s we have Z t Z +∞ u(t) = U (t, ξ)P (ξ)f (ξ)dξ − U (t, ξ)Q(ξ)f (ξ)dξ −∞

= U (t, s)

t

Z

s

−∞

U (s, ξ)P (ξ)f (ξ)dξ −

Z

+∞

U (s, ξ)Q(ξ)f (ξ)dξ s

Z t U (t, ξ)(Q(ξ)f (ξ)dξ U (t, ξ)P (ξ)f (ξ)dξ + s s Z t = U (t, s)u(s) + U (t, ξ)f (ξ)dξ. +

Z

t

s

On the other hand, we have

ku(t)k = k ≤ ≤

Z

Z

+∞

G(t, s)f (s)dsk

−∞ +∞

−∞ Z +∞ −∞

Ke−α|t−s| kf (s)kds Ke−α|η| kf (t − η)kdη.

An easy computation shows that as f ∈ C0 (R, X), the above estimate yields u ∈ C0 (R, X). Now we prove the implication ”(ii) ⇒ (i)”. First, we notice that by the above lemma and the Closed Graph Theorem (ii) yields that there exists a positive constant k such that kL−1 f k ≤ ckf k,

∀f ∈ C0 (R, X).

(2.4)

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To proceed we need several lemmas: Lemma 2.2. Let (ii) hold. Then there exist positive constants M, β > 0 depending only on c such that kU (t, t0 )xk ≤ M e−β(t−s) kU (s, t0 )xk,

∀t ≥ s ≥ t0 ,

(2.5)

provided that lim U (t, t0 )x = 0.

t→+∞

(2.6)

Proof. Without loss of generality we may assume that x 6= 0. Set u(t) := U (t, t0 )x, ∀t ≥ t0 and t1 := sup{t ≥ t0 : u(t) 6= 0}. Obviously, by definition, u(t) 6= 0 for all t ∈ [t0 , t1 ) and u(t) = 0 for all t ≥ t1 . Next, we define a smooth function φn (t) for every n ∈ N with compact suppφn in [t0 , t1 ) such that 0 ≤ φn (t) ≤ 1, φn (t) = 1 for all t ∈ [t0 + 1/n, min(n, t1 − 1/n)]. Set φn (t) u(t), ku(t)k

f (t) =

t ∈ R.

It is clear that f is a well-defined continuous function on R and f ∈ C0 (R, X). Consider the function Z t φn (s) xf (t) := u(t) ds, t ≥ t0 . t0 ku(s)k

By the definition of φn we see that xf can be naturally extended to the whole line as a continuous function by setting xf (t) = 0 for all t 6∈ [t0 , min(n, t1 )]. Consequently, we have Z s Z t xf (t) = U (t, s)u(s) φn (ξ)ku(ξ)k−1 dξ + U (t, ξ)u(ξ)φn (ξ)ku(ξ)k−1 dξ t0

= U (t, s)xf (s) +

s

Z

t

U (t, ξ)f (ξ)dξ, s

∀ t ≥ s,

i.e., xf ∈ C0 (R, X) is a solution to Eq. (2.1). By assumption and (2.4) we have kxf k ≤ ckf k, so, since kf k = 1, ku(t)k Letting n → ∞ we get ku(t)k

t

Z

t0

Z

t

t0

φn (s) ds ≤ c, ∀t ≥ t0 . ku(s)k

ku(s)k−1 ds ≤ c, ∀t ≥ t0 .

(2.7)

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It is easy to see that the above inequality and the bounded growth of the process yield that there are positive constants K, β independent of x, t0 such that kU (t, t0 )xk = u(t) ≤ Ke−β(t−s) u(s) = Ke−β(t−s) U (s, t0 ), ∀t ≥ s ≥ t0 . (2.8) Rt In fact, set g(t) = t0 ku(s)k−1 ds. Then by (2.7) we obtain 1 d ln g(t) ≥ , ∀t > t0 . dt c Next, assuming that τ > 1 is given, integrating both sides of the above over the interval [t0 + 1, t0 + τ ) we get τ −1 ln g(t0 + τ ) − ln g(t0 + 1) ≥ . c Hence, τ −1 g(t0 + τ ) ≥ g(t0 + 1)e c . (2.9) ω(t−s) Now, by the bounded growth, i.e., kU (t, s)k ≤ N e , ∀t ≥ s, for any s ∈ [t0 , t0 + 1] we have ku(s)k ≤ ku(t0 )kN eω . Consequently, without loss of generality assuming that t1 > t0 + 1 we have Z t0 +1 1 (2.10) g(t0 + 1) = ku(s)k−1 ds ≥ e−ω ku(t0 )k−1 . N t0 By (2.7), for τ > 1 we get c ku(t0 + τ )k ≤ . g(t0 + τ ) By (2.9), ce−(τ −1)c c ≤ . g(t0 + τ ) g(t0 + 1) Next, by (2.10) ce−(τ −1)c ≤ N eω ku(t0 )kce−(τ −1)c = N1 e−τ /cku(t0 )k g(t0 + 1) where N1 = cN e1/c+ω . Thus for τ > 1 we have ku(t0 + τ )k ≤ N1 e−τ /c ku(t0 )k (2.11) On the other hand, for 0 ≤ τ ≤ 1 using the bounded growth we have ku(t0 + τ )k ≤ N e1/c+ω e−τ /cku(t0 )k . (2.12) Finally, combining (2.12) with (2.11) we have ku(t0 + τ )k ≤ N2 e−τ /cku(t0 )k, ∀0 < τ < t1 , where N2 := max{N1 , N e1/c+ω }. This finishes the proof.

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Corollary 2.1. Let (ii) hold. Then X1 (t0 ) := {x ∈ X : lim U (t, t0 )x = 0} t→+∞

(2.13)

is a closed subspace of X. Proof.

This corollary is an immediate consequence of the above lemma.

To proceed we need a notion of extension. Definition 2.3. We say that the function u(t) := U (t, t0 )x for t ≥ t0 is extendable to a function on the whole line as a solution of the corresponding homogeneous equation (also denoted by u(·)), if u is extendable to the whole line such that u(t) = U (t, s)u(s), ∀t ≥ s, t, s ∈ R. Note that if (ii) holds, then u(t) has at most one extension on (−∞, t0 ] such that it vanishes at infinity. In fact if we have two extensions, then by denoting by u1 (t) and u2 (t) the two functions defined on the whole line, respectively, we see that ( 0, ∀t ≥ t0 w(t) = u1 (t) − u2 (t), ∀t ≤ t0 is a solution in C0 (R, X) to (2.1) with f = 0. Hence, by (ii), u1 (t) = u2 (t), ∀t ≤ t0 . Lemma 2.3. Let (ii) hold and let u(t) := U (t, t0 )x. Then there exist positive constants K 0 , β 0 independent of x such that 0

ku(t)k ≤ K 0 e−β (s−t) ku(s)k,

∀t ≤ s ≤ t0 ,

(2.14)

provided that u has an extension to the whole line as a solution of the corresponding homogeneous equation vanishing at −∞. Proof. Suppose that u(t) := U (t, t0 )x, t ≥ t0 can be extended to the whole line as a solution to Eq. (2.1). Consider the function f (t) = −

ϕn (t) u(t), ku(t)k

t ∈ R,

where for every n ∈ N    1, ∀2/n ≤ t ≤ t0 + n, ϕn (t) := 1 − (t − t0 − n), ∀t0 + n ≤ t ≤ t0 + n + 1   0, ∀t ≥ t + n + 1, or t ≤ 1/n. 0

(2.15)

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Clearly that y(t) = u(t)

Z

∞

ϕn (s)ku(s)k−1 ds

t

is a solution of (2.1) with f defined by (2.15). By the same reasoning as above and by letting n to infinity we can show that Z ∞ ku(t)k ku(s)k−1 ds ≤ c, t

where c is defined by (2.4). Finally, as above we can prove (2.14).

A consequence of this lemma is the following: Corollary 2.2. Let (ii) hold. Then X2 (t0 ) := {x ∈ X : u(t) := U (t, t0 )x has an extension on (−∞, t0 ] as a solution to the homogeneous equation vanishing at −∞} is a closed subspace of X. Lemma 2.4. Let (ii) hold. Then for every t0 ∈ R X = X1 (t0 ) ⊕ X2 (t0 ),

(2.16)

where X1 (t0 ) and X2 (t0 ) are defined by (2.13) and (2.16). Proof. For every x ∈ X let φ(t) be a continuously differentiable function on R with the following properties: suppφ ⊂ [t0 , +∞), φ(t) = 1, ∀t ≥ t0 + 1 and |φ0 (t)| ≤ 2. Next, let f (t) = φ0 (t)U (t, t0 )x. Obviously, f can be continuously extended to the whole line by setting f (t) = 0, ∀t ≤ t0 . By the assumption there exists a unique solution xf ∈ C0 (R, X) to (2.1). The function v(t) := φ(t)U (t, t0 )x, t ≥ t0 is also a solution on [t0 , +∞) of (2.1). Set z(t) := v(t)−xf (t) and y(t) = U (t, t0 )x−z(t) for t ≥ t0 . Then obviously z(t) and y(t) are solutions of the corresponding homogeneous equation of (2.1) on (−∞, +∞) and [t0 , ∞), respectively. By definition, we have lim kz(t)k = 0,

t→−∞

lim ky(t)k = 0

t→+∞

that is, z(t0 ) = −xf (t0 ) ∈ X2 (t0 ) and y(t0 ) = x − z(t0 ) ∈ X1 (t0 ). This proves the lemma. Now we finish the proof of the theorem by choosing the projection Q(t) as the projection of X = X1 (t) ⊕ X2 (t) onto X2 (t) for every t ∈ R.

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Remark 2.1. If the evolutionary process (U (t, s))t≥s is determined by a C0 -semigroup T ((t))t≥0 that is generated by an operator A, then the condition σ(A) ∩ iR = ∅ does not guarantee the hyperbolicity of T ((t))t≥0 , so it does not imply the exponential dichotomy of (U (t, s))t≥s . This is because in general the Spectral Mapping Theorem does not hold for arbitrary C0 -semigroup. As a counterexample we can take the semigroup (T (t))t≥0 in Example 1.12. Obviously, since σ(A) = ∅, we have σ(A) ∩ iR = ∅ and kT (t)k = 1 for all t ≥ 0. Hence, if (T (t))t≥0 is hyperbolic, then the projection Q in the Definition 2.2 must be trivial, i.e., Q = I. That means kT (t)k ≤ Ke−αt for all t ≥ 0, where K, α are some positive numbers. But this contradicts to the fact that kT (t)k = 1 for all t ≥ 0. 2.2

Evolution Semigroups and Perron Theorem

In this section we will discuss the relation between the Perron Theorem and the so-called evolution semigroup of bounded linear operators on C0 (R, X) associated with a given evolutionary process. Definition 2.4. Let (U (t, s))t≥s be a strongly continuous evolutionary process on R in a Banach space X. Then the following family (T h )h≥0 of linear operators on C0 (R, X) is called the evolution semigroup associated with (U (t, s))t≥s on the function space C0 (R, X): h T v (t) = U (t, t − h)v(t − h), ∀v ∈ C0 (R, X), h ≥ 0, t ∈ R. (2.17)

Lemma 2.5. (T h )h≥0 is a strongly continuous semigroup of linear operators on C0 (R, X) with the infinitesimal generator G such that D(G) = D(L) and Gu = −Lu for all u ∈ D(L).

Proof. First, we show that for every v ∈ C0 (R, X) with compact support we have lim T h v = v. h↓0

Since suppv is compact, the range R(v) of v is compact. By the strong continuity of the process (U (t, s))t≥s we have that the map ∆ × X 3 (t, s, x) 7→ U (t, s)x ∈ X is continuous (here ∆ := {(t, s) ∈ R2 : t ≥ s}. Thus it is uniformly continuous on (∆ ∩ [0, t0 ]2 ) × R(v). On the other hand, v is uniformly continuous on R. Thus, for every ε > 0 there exists a positive δ such that for |h| < δ kT h v − vk = sup kU (t, t − h)v(t − h) − v(t)k < ε. t∈R

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Obviously, since there are positive constants M, ω such that kU (t, s)k ≥ M eω(t−s) for all t ≥ s, we have kT h vk ≤ M eh kvk, ∀h ≥ 0. Thus, for arbitrary w ∈ C0 (R, X) there is a sequence vn with compact support in C0 (R, X)) convergent to w as n → ∞. We have 0 ≤ lim sup kT h w − wk ≤ (1 + M eω )kvn − wk. h↓0

This shows that limh↓0 T h w = w, i.e., the strong continuity of the evolution semigroup. Now we suppose that u ∈ D(L). Then there is a unique f ∈ C0 (R, X) such that (2.1) holds. We have h 1 T u−u (t) = (U (t, t − h)u(t − h) − u(t)) h h Z 1 t =− U (t, ξ)f (ξ)dξ. h t−h Since f ∈ C0 (R, X), f is uniformly continuous and R(f ) is relatively compact. Consequently, (t, s, x) 7→ U (t, s)x is uniformly continuous on (∆ ∩ [0, t0 ]2 ) × R(f ). This yields that Z 1 t kU (t, ξ)f (ξ) − f (ξ)kdξ = 0, lim sup h↓0 t∈R h t−h i.e., h T u−u lim = −f. (2.18) h↓0 h Conversely, let u ∈ D(G). By the basic properties of C0 -semigroups we have that the map (0, ∞) 3 h 7→ T h u is differentiable and d h T u = Gu, ∀h > 0 dh and Z h h T u−u= T ξ Gudξ, ∀h ≥ 0. 0

Setting f := Gu we have Z h h T u − u (t) = U (t, t − h)u(t − h) − u(t) = [[T ξf ] (t)dξ 0 Z t = U (t, ξ)f (ξ)dξ. t−h

Hence,

u(t) = U (t, t − h)u(t − h) −

Z

t

U (t, ξ)f (ξ)dξ, t−h

∀t ∈ R, h ≥ 0.

This shows that u is a solution of (2.1) with the forcing term −f and finishes the proof of the lemma.

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Remark 2.2. It is easy to see that if U (t, s) = e(t−s)A :=

∞ X ((t − s)A)k k=0

k!

,

where A ∈ L(X), then D(G) = {u ∈ C0 (R, X) : ∃u0 ∈ C0 (R, X)} and G is of the form d Gu(t) = − u(t) + Au(t), ∀u ∈ D(G). dt Let U (t, s) = T (t − s) for certain C0 -semigroup (T (t))t≥0 with generator A. For unbounded A, the above formula is no longer true. To see how it should be modified we consider the evolution semigroup (T h )h≥0 as the product of two commutative semigroups (R(h))h≥0 and (S(h))h≥0 defined as follows [R(h)v](t) = T (h)v(t);

[S(h)v](t) = v(t − h),

∀h ≥ 0, v ∈ C0 (R, X).

Let A be the operator with D(A) = {u ∈ C0 (R, X) : u(t) ∈ D(A), ∀t ∈ R, Au(·) ∈ C0 (R, X)}, and Au = Au(t), ∀t ∈ R, u ∈ D(A). Clearly that the infinitesimal generator of (S(t))t∈R is −d/dt with D(−d/dt) = {f ∈ C0 (R, X) : ∃f 0 ∈ C0 (R, X)}. Thus, by Exercise 7, we have G = −d/dt + A. As an immediate consequence of the above lemma we have Corollary 2.3. The process (U (t, s))t≥s has an exponential dichotomy if and only if the generator G of the evolution semigroup (T h )h≥0 associated with (U (t, s))t≥s is invertible. Corollary 2.4. If λ ∈ σ(G), then λ + iµ ∈ σ(G) for all µ ∈ R. Proof. Let Uµ (t, s) := e−iµ(t−s) U (t, s), for fixed µ ∈ R. Note that (U (t, s))t≥s has an exponential dichotomy if and only if so does (Uµ (t, s))t≥s . Obviously, the generator of the evolution semigroup associated with (Uµ (t, s))t≥s is the operator G − iµI with the same domain. By the above corollary, G is invertible if and only if so is G − iµI. Definition 2.5. Let (T (t))t≥0 be a C0 -semigroup on X. It is said to be hyperbolic if σ(T (1))∩Γ = ∅, where Γ denotes the unit circle on the complex plane.

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Theorem 2.2. Let (U (t, s))t≥s be a strongly continuous evolutionary process and let (T h )h≥0 be its evolution semigroup associated with it on C0 (R, X). Then (U (t, s))t≥s has an exponential dichotomy if and only if (T h )h≥0 is hyperbolic. Proof. By the inclusion of C0 -semigroups, if σ(T 1 ) ∩ Γ = ∅, then 0 6∈ σ(G). By the above corollaries (U (t, s))t≥s has an exponential dichotomy. Conversely, if (U (t, s))t≥s has an exponential dichotomy, then we set C1 := {u ∈ C0 (R, X) : u(t) ∈ ImP (t), ∀t ∈ R} and C2 := {u ∈ C0 (R, X) : u(t) ∈ ImQ(t), ∀t ∈ R}, where projections P (t), Q(t), t ∈ R are determined from the exponential dichotomy of (U (t, s))t≥s . It is easy to see that T 1 is strictly contractive on C1 and is left C2 invariant. Moreover, the restriction of T 1 to C2 is invertible and its inverse is also strictly contractive. By the Spectral Radius Theorem σ(T 1 ) ∩ Γ = ∅. The following spectral mapping theorem holds for evolution semigroups. Theorem 2.3. We have the spectral mapping theorem for the evolution semigroup (T h )h≥0 σ(T h )\{0} = ehσ(G) ,

∀h ≥ 0.

(2.19)

Proof. By the above theorem 1 ∈ ρ(T 1 ) if and only if Γ ⊂ ρ(T 1 ). For any positive µ ∈ R+ consider the process Vµ (t, s) := e−µ(t−s) U (t, s), applying the above theorem to this process we have that µ ∈ ρ(T 1 ) if and only if the whole circle {|z| = µ} is contained in ρ(T 1 ). Since 0 ∈ ρ(G) if and only if 1 ∈ ρ(T 1 ), by Lemma 2.4 we have σ(T 1 )\{0} = eσ(G) .

For arbitrary η > 0 we consider the process Vh (t, s) := e−η(t−s) U (t, s). Its evolution semigroup is Tηh )h≥0 defined by Tηh v(t) = e−η T h . Hence, 1 ∈ ρ(Tη1 ) = eη ρ(T 1 ) if and only if Γ ⊂ ρ(Tη1 ) = eη ρ(T 1 ). Repeating the above reasoning we have (2.19). Next, we consider the perturbation theory of exponential dichotomy of processes. Let (U (t, s))t≥s be a strongly continuous evolutionary process and let B : R → L(X) be continuous. Then we can associated with the following integral equation Z t x(t) = U (t, s)x(s) + U (t, ξ)B(ξ)x(ξ)dξ, ∀t ≥ s (2.20) s

a strongly continuous evolutionary process (V (t, s))t≥s as follows: for any t ≥ s we define V (t, s)y as the unique solution to the above equation

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such that x(s) = y. We can show without difficulty that such a family of operators forms a strongly continuous evolutionary process. Moreover, denoting its evolutionary process by (TBh )t≥s we can prove that kTB1 − T 1 k ≤ supt∈R kB(t)k. Hence, we have the following Corollary 2.5. Let (U (t, s))t≥s have an exponential dichotomy. Then the process (V (t, s))t≥s defined as above has an exponential dichotomy as well provided that supt∈R kB(t)k is sufficiently small. Proof. By the above theory, 1 ∈ ρ(T 1 ). For sufficiently small 1 supt∈R kB(t)k we have 1 ∈ ρ(TB ). This yields the hyperbolicity of TBh , and hence the exponential dichotomy of (V (t, s))t≥s follows. Example 2.1. If U (t, s) = T (t − s), where (T (t))t≥0 is a C0 -semigroup, then we can easily show that (U (t, s))t≥s has an exponential dichotomy if and only if T (1) is hyperbolic. In fact, in this case we consider the Riesz projection Z 1 P = (λ − T (1))−1 dλ, 2πi Γ

where Γ is positively oriented. By the general theory of linear operators X = ImP ⊕ KerP and, σ(T (1)) = σ(T (1)|ImP ) t σ(T (1)|KerP ). Hence, by the Spectral Radius Theorem there are positive constants K, α such that kT (t)xk ≤ Ke−αt kxk,

kT (s)yk ≤ Ke

−αs

kyk,

∀t ≥ 0; x ∈ ImP

∀s ≤ 0; y ∈ KerP,

where T (s)|KerP is the inverse of T (−s)|KerP for s > 0 whose existence is guaranteed by the above decomposition of spectrum of T (1). Thus, we can check that the process U (t, s) = T (t − s) has an exponential dichotomy. If (T (t))t≥0 is eventually norm continuous with generator A, then by Spectral Mapping Theorem, for (T (t))t≥0 to have an exponential dichotomy it is necessary and sufficient that σ(A) ∩ iR = ∅. 2.3 2.3.1

Stability Theory Exponential Stability

In this subsection we consider the exponential stability of a strongly continuous evolutionary process (U (t, s))t≥s with t, s ∈ R+ .

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Definition 2.6. An evolutionary process (U (t, s))t≥s is said to be exponentially stable if there are positive constants K, α such that kU (t, s)k ≤ Ke−α(t−s) ,

∀t ≥ s.

The main result of this subsection is the following Theorem 2.4. Let (U (t, s))t≥s with t ≥ s ≥ 0 be a strongly continuous evolutionary process. Then, (U (t, s))t≥s is exponentially stable if and only if for every f ∈ C0 (R, X) the following function Z t U (t, ξ)f (ξ)dξ, t ∈ R+ (2.21) uf (t) = 0

is in C0 (R, X). Proof.

For every t ≥ 0 we define an operator V (t) ∈ L(C0 (R, X), X) Z t V (t) : C0 (R, X) 3 f 7→ U (t, ξ)f (ξ)dξ ∈ X. 0

By the assumption, for every f ∈ C0 (R, X) we have sup kV (t)f k < ∞. t≥0

Thus, by the Uniform Boundedness Principle, sup kV (t)k < ∞. t≥0

Consequently, there is a positive constant c > 0 such that kuf k ≤ ckf k,

f ∈ C0 (R, X).

By the same reasoning as in the proof of Lemma 2.2 we can show that there are positive constants K, α depending only on c such that kU (t, s)k ≤ Ke−α(t−s) ,

This finishes the proof of the theorem.

∀t ≥ s ≥ 0.

We are now interested in the stability of individual orbits in the case the process is not exponentially stable. To this end we introduce an operator G : D(G) ⊂ BC(R+ , X) → BC(R+ , X), where we denote by BC(R+ , X) the space of all X valued bounded and continuous functions on R+ with sup-norm as follows: D(G) := {u ∈ BC(R+ , X) such that ∃f ∈ BC(R+ , X) Z t U (tξ)f (ξ)dξ, ∀t ∈ R+ } u(t) = 0

Gu := f,

∀u ∈ D(G).

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As for the operator L in the previous section we can show that G is closed. Theorem 2.5. Assume that there is a positive constant ν > 0 such that kGuk ≥ νkuk,

∀u ∈ D(G).

(2.22)

Then there exist positive constants K, α depending only on ν such that kU (t, t0 )xk ≤ Ke−α(t−s) kU (s, t0 )xk,

provided that

∀t ≥ s ≥ t0

sup kU (t, t0 )xk < ∞. t≥0

Proof. The proof can be done in the same manner as in Lemma 2.2, so the details are omitted. Now we consider the perturbation theory of an exponentially stable process (U (t, s))t≥s . As in the previous section for a given bounded B : R+ → L(X) we can show that the following integral equation Z t x(t) = U (t, s)x(s) + U (t, ξ)B(ξ)x(ξ)dξ, ∀t ≥ s ≥ 0 s

generates a strongly continuous evolutionary process (V (t, s))t≥s . Using the Gronwall inequality it is easy to prove the following assertion:

Proposition 2.1. Let (U (t, s))t≥s be an exponentially stable strongly continuous evolutionary process. Then the process (V (t, s))t≥s is also exponentially stable provided that supt≥0 kB(t)k is sufficiently small. Proof.

Let ε := sup kB(t)k t≥0

and η(t) := V (t, s)x for any fixed s ∈ R+ , x ∈ X. Then Z t −α(t−s) η(t) ≤ Ke kxk + εKe−α(t−ξ) η(ξ)dξ, s

∀t ≥ s.

Applying the Gronwall inequality to the function ζ(t) := eαt η(t) we get

Thus Therefore, if

η(t) ≤ Ke−(α−Kε)(t−s) kxk, kV (t, s)k ≤ Ke−(α−Kε)(t−s) , α K is exponentially stable.

∀t ≥ s. ∀t ≥ s.

ε<

then (V (t, s))t≥s

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2.3.2

Strong Stability

In this subsection we will prove the following theorem: if (T (t))t≥0 is a uniformly bounded C0 -semigroup such that the unitary spectrum σ(A)∩iR is at most countable and σp (A∗ ) ∩ iR = ∅, then (T (t))t≥0 is strongly stable, i.e. limt→∞ kT (t)xk = 0 for all x ∈ X. Definition 2.7. A bounded operator T on a Banach space X is called an isometry if kT xk = kxk for all x ∈ X. Lemma 2.6. Let (T (t))t≥0 be a C0 -semigroup of isometries on a Banach space X, and let A be its generator. Then the following hold: (i) For all x ∈ D(A) and λ ∈ C we have k(λ − A)xk ≥ |<λ| kxk; (ii) If E ⊂ R is closed and x ∈ X is such that the map λ 7→ R(λ, A)x has a holomorphic extension F to a connected neighbourhood V of {<λ ≥ 0}\iE, then for all λ ∈ V \iR we have kF (λ)k ≤ |<λ|−1 kxk. Proof.

(i): First, we may assume that <λ 6= 0. From the identity Z t e−λt T (t)x = x + e−λs (A − λ)T (s)x ds 0

we have

e−Re λt kxk = e−Re λt kT (t)xk Z t ≤ kxk + e−Re λs kT (s)(λ − A)xk ds 0 Z t = kxk + e−Re λs ds k(λ − A)xk 0

e−Re λt − 1 k(λ − A)xk. = kxk + −Re λ This proves the lemma for <λ < 0. For <λ > 0 the inequality follows from the Laplace transform representation of the resolvent. (ii): This is proved in the same way, after first substituting R(λ, A)x for x in the first formula and passing to the holomorphic extension. Theorem 2.6. Let (T (t))t≥0 be C0 -semigroup of contractions on a Banach space X, with generator A. Then there exists a Banach space Y , a bounded operator π : X → Y with dense range, and a C0 -semigroup (U (t))t≥0 of isometries on Y with generator B such that: (i) U (t)π = πT (t) for all t ≥ 0. Moreover, πD(A) ⊂ D(B) and Bπx = πAx for all x ∈ D(A);

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(ii) limt→∞ kT (t)xk = kπxk for all x ∈ X; (iii) σ(B) ⊂ σ(A). If σ(A) ∩ iR is a proper subset of iR, then (U (t))t≥0 extends to a C0 -group of isometries. Proof. On X we define the seminorm l by l(x) := limt→∞ kT (t)xk. Since (T (t))t≥0 is contractive, this limit indeed exists. Let π : X → Y0 := X/ker l be the quotient mapping. The seminorm l induces a norm l0 on Y0 by l0 (πx) := l(x), and hence l0 (πx) = l(x) = lim kT (t)xk. t→∞

For t ≥ 0, we define U0 (t) : Y0 → Y0 by U0 (t)πx := πT (t)x. We have l0 (U0 (t)πx) = l0 (πT (t)x) = l(T (t)x) = l(x) = l0 (πx). This shows that U0 (t) is isometric with respect to the norm l0 . Let Y be the completion of Y0 with respect to l0 . Then each operator U0 (t) extends to an isometry U (t) on Y . Strong continuity of the family (U (t))t≥0 follows from the density of πX in Y , the contractivity of the operators U (t), and the estimate lim sup kU (t)πx − πxk = lim sup t↓0

t↓0

lim kT (t + s)x − T (s)xk

s→∞

≤ lim sup kT (t)x − xk = 0, t↓0

x ∈ X.

If x ∈ D(A), then

1 1 lim (U (t)πx − πx) = π lim (T (t)x − x) = πAx, t↓0 t t↓0 t

proving that πx ∈ D(B) and Bπx = πAx. We have proved (i) and (ii). Next, we prove (iii). Let λ ∈ %(A). We define a linear operator Rλ on Y0 as follows Rλ πx := πR(λ, A)x. This operator is well-defined and l0 (Rλ πx) = lim kT (t)R(λ, A)xk t→∞

≤ kR(λ, A)k lim kT (t)xk = kRk l0 (πx). t→∞

Therefore, Rλ extends to a bounded operator on Y and kRλ k ≤ kR(λ, A)k. For all x ∈ X we have Rλ πx = πR(λ, A)x ∈ πD(A) ⊂ D(B) and (λ −

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B)Rλ πx = (λ−B)πR(λ, A)x = π(λ−A)Rx = x. Similarly, for all x ∈ D(A) we have πx ∈ D(B) and Rλ (λ − B)πx = Rλ π(λ − A)x = πR(λ − A)x = x. Therefore, to prove that Rλ is a two-sided inverse of λ − B, in view of the closedness of B it remains to prove that πD(A) is dense in D(B) with respect to the graph norm. So let y ∈ D(B) be arbitrary. Fix µ > 0 arbitrary and choose z ∈ Y such that y = R(µ, B)z. Take a sequence {xn } ⊂ X such that πxn → z in Y and put yn := πR(µ, A)xn . Then yn ∈ πD(A), yn = R(µ, B)πxn → R(µ, B)z = y, and Byn = BR(µ, B)πxn → BR(µ, B)z = By. Here we used that πR(µ, A) = R(µ, B)π by the Laplace transform representation of the resolvents and that BR(µ, B) = µR(µ, B)− I is a bounded operator on Y . Thus, yn → y in D(B) with respect to the graph norm. This concludes the proof of (iii). Suppose σ(A) ∩ iR is properly contained in iR. We have to prove that (U (t))t≥0 extends to a C0 -group of isometries. By Lemma 2.6 (i), for all y ∈ D(B) and <λ < 0 we have k(λ − B)yk ≥ |<λ| kyk. It follows that the open left half-plane C− contains no approximate eigenvalues for B. In particular, C− contains no elements of the boundary of σ(B). Hence, either C− ⊂ σ(B) or C− ∩ σ(B) = ∅. But in the first case, also iR ⊂ σ(B) since σ(B) is closed. This contradicts the assumption, so we must have the second alternative. It follows that σ(B) ⊂ iR. For <λ > 0 we have kR(λ, −B)k = kR(−λ, B)k ≤

1 1 = . |<(−λ)| <λ

By the Hille-Yosida theorem, −B is the generator of a C0 -semigroup (V (t))t≥0 of contractions on Y . We check that U (t) is invertible for all t ≥ 0 with inverse V (t). For all x ∈ D(B) = D(−B), the maps t 7→ U (t)V (t)x and t 7→ V (t)U (t)x are differentiable with derivative identically zero. It follows that the maps are constant, and by letting t ↓ 0 it follows that U (t)V (t) = V (t)U (t) = I on the dense set D(B), hence on all of Y . Finally, each operator V (t) is an isometry, being the inverse of an isometry. The triple (Y, π, (U (t))t≥0 ) will be called the isometric limit (semi)group associated to (T (t))t≥0 . Corollary 2.6. If (T (t))t≥0 is an isometric C0 -semigroup on X with σ(A)∩ iR 6= iR, then (T (t))t≥0 extends to an isometric C0 -group.

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Proof. By the isometric nature of (T (t))t≥0 we have l(x) = x for all x ∈ X, so Y = X and (T (t))t≥0 = (U (t))t≥0 . But (U (t))t≥0 extends to a C0 -group since σ(A) ∩ iR 6= iR. An invertible operator is called doubly power bounded if sup kT k k < ∞. k∈Z

Lemma 2.7. Let T be a doubly power bounded operator on a Banach space X with σ(T ) = {1}. Then T = I. Proof. Since ln z is holomorphic in a neighbourhood of z = 1, by Dunford calculus we may define the bounded operator S := −i ln T. Then T = eiS and the spectral mapping theorem implies that σ(mS) = {0} for all m ∈ N. Also, for all m ∈ N we have σ(sin(mS)) = sin(σ(mS)) = {sin 0} = {0}, and

m

T − T −m n n

≤ sup kT k k.

k(sin(mS)) k =

k∈Z 2i P∞ Let n=0 cn z n be Taylor series of the principle branch of arcsin z at z = P∞ π 0. As is well-known, cn ≥ 0 for all n and n=0 cn = arcsin(1) = 2 . Consequently, kmSk = k arcsin(sin(mS))k ≤

∞ X

n=0

cn k(sin(mS))n k ≤

π sup kT k k. 2 k∈Z

Since this holds for all m ∈ N, it follows that S = 0 and T = eiS = I.

Theorem 2.7. Let (T (t))t≥0 be a uniformly bounded C0 -semigroup on a Banach space X, with generator A. If (i) σ(A) ∩ iR is countable, and (ii) σp (A∗ ) ∩ iR = ∅, then (T (t))t≥0 is strongly stable, i.e. limt→∞ kT (t)xk = 0 for all x ∈ X. Proof. By renorming with the equivalent norm kxk∗ := supt≥0 kT (t)xk, we may assume that (T (t))t≥0 is contractive. Let (Y, π, (U (t))t≥0 ) be the isometric limit semigroup associated to (T (t))t≥0 , and let B be the generator of (U (t))t≥0 . By (i), σ(A) ∩ iR cannot be all of iR, and therefore (U (t))t≥0 extends to an isometric group on Y . Assuming that (T (t))t≥0 is not strongly stable, we shall prove that (i) implies σp (A∗ ) ∩ iR 6= ∅. In fact, since (T (t))t≥0 is not strongly stable, the definition of Y implies that Y 6= {0}. By Lemma 1.1, σ(B) 6= ∅. Also, since σ(B) ⊂ σ(A),

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it follows that σ(B) is countable. In particular, since σ(B) is closed and countable, it contains an isolated point, say iω. Let Pω be the associated spectral projection in Y , let (Uω (t))t≥0 be the restriction of (U (t))t≥0 to Pω Y and let Bω denote its generator. Since σ(Bω ) = {iω}, Theorem 1.10 implies that σ(Uω (t)) = {eiωt } for all t ∈ R. By Lemma 2.7, this implies that Uω (t) = eiωt I. Hence, for all y ∈ Y we have Uω (t)Pω y = eiωt Pω y, so Pω y ∈ D(Bω ) and Bω Pω y = iωPω y. Fix an arbitrary non-zero yω∗ ∈ (Pω Y )∗ and define x∗ ∈ X∗ by < x∗ , x >:=< yω∗ , Pω πx >,

x ∈ X.

Then x∗ 6= 0. For all x ∈ D(A) we have πx ∈ D(B), Bπx = πAx, and < x∗ , Ax > = < yω∗ , Pω πAx >=< yω∗ , Pω Bπx >

= < yω∗ , Bω Pω πx >= iω < yω∗ , Pω πx >= iω < x∗ , x > . Hence, x∗ ∈ D(A∗ ) and A∗ x∗ = iωx∗ . 2.4 2.4.1

Comments and Further Reading Guide Further Reading Guide

So far we have discussed conditions for the stability and exponential dichotomy of homogeneous equations. Below we state without proofs further important results on the asymptotic behavior of C0 -semigroups. The reader is referred to Chap. 5 §7 in [van Neerven (78)]) for more information. Definition 2.8. A C0 -semigroup (T (t))t≥0 on X is said to be almost periodic if for each x ∈ X the set {T (t)x, t ∈ [0, +∞)} is relatively compact in X. Theorem 2.8. (Theorem 5.7.10 in [van Neerven (78)]) Let (T (t))t≥0 be a uniformly bounded C0 -semigroup on a Banach space X, with generator A, and assume that σ(A) ∩ iR is countable. Then the following assertions are equivalent: (1) (T (t))t≥0 is almost periodic, Rt (2) For every iω ∈ σ(A) ∩ iR the limit limt→∞ 1t 0 e−iωs T (s)x ds exists for every x ∈ X, (3) For every iω ∈ σ(A) ∩ iR, R(A − iω) + N (A − iω) is dense in X. The following is referred to as the splitting Theorem of Glicksberg and DeLeeuw.

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Theorem 2.9. (Theorem 5.7.7 in [van Neerven (78)]) Let (T (t))t≥0 be an almost periodic C0 -semigroup on a Banach space X. Then there exists a direct sum decomposition X = X0 ⊕ X1 of (T (t))t≥0 -invariant subspaces, where X0 = {x ∈ X : lim kT (t)xk = 0} t→∞

and X1 is the closed linear span of all eigenvectors of the generator A with purely imaginary eigenvalues. Moreover, the restriction of (T (t)) t≥0 to X1 extends to an almost periodic C0 -group on X1 . If (T (t))t≥0 is contractive, this group is isometric. Exponential dichotomy of homogeneous equations that are defined on the half line is more difficult for us to study. Evolution semigroups method actually fails. However, Perron type characterization of exponential dichotomy can be used for these equations. We refer the reader to [Minh, R¨ abiger and Schnaubelt (70)] for more details in this direction. Nevertheless, evolution semigroups method is a strong tool to study the exponential dichotomy of linear skew products. We refer the reader to [Chicone and Latushkin (19)] for more details in this direction. New applications of evolution semigroups to the stability problem of homogeneous equations can be found in [Batty, Chill and Tomilov (15); Latushkin and Tomilov (56)]. The explicit formula for the generators of evolution semigroups gives rise to new applications of this method to the study of inhomogeneous linear and semilinear equations. This is the content of the next chapter. 2.4.2

Comments

Perron Theorem for evolutionary processes in Banach spaces was first proved by Zhikov (see [Zikov (108)]). The proof that is given in Section 1 is an adaptation of this proof for the function space C0 (R, X). Another recent proof of the characterization of exponential dichotomy using evolution semigroups was given by Latushkin and Montgomery-Smith in [Latushkin, Monthomery-Smith (55)] (see also [Chicone and Latushkin (19); van Neerven (78)]). The explicit formula for the generators of evolution semigroups was found in [Aulbach and Minh (10)] for general semilinear evolution equations. In the linear case, it was also found in [Baskakov (13)]. The presentation of Section 2 follows [Minh (69)]. The Perron condition (that is similar to the condition in Theorem 2.4) for exponential stability was first proved by Datko [Datko (25)]. The

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proof of Theorem 2.4 is taken from [Minh, R¨ abiger and Schnaubelt (70)]. Theorem 2.7, that is widely referred to as ABLV Theorem, was first proved by Sklyar and Shirman for the bounded case [Skylar and Shirman (97)] (we thank G.M. Sklyar for sending us their original paper). It is amazing that their result and method of proving can be extended to the unbounded case by Lyubich and Vu in [Lyubich and Vu (66)]. The ABLV Theorem was proved independently by Arendt and Batty in [Arendt and Batty (4)].

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Chapter 3

Existence of Almost Periodic Solutions to Inhomogeneous Equations The problem of our primary concern in this chapter is to find spectral conditions for the existence of almost periodic solutions of periodic equations. Although the theory for periodic equations can be carried out similarly to that for autonomous equations, there is always a difference between them. This is because in general there is no Floquet representation for the monodromy operators in the infinite dimensional case. Section 1 will deal with evolution semigroups acting on invariant function spaces of AP (X). Since, originally, this technique is intended for nonautonomous equations we will treat equations with as much nonautonomousness as possible, namely, periodic equations. The spectral conditions are found in terms of spectral properties of the monodromy operators. Meanwhile, for the case of autonomous equations these conditions will be stated in terms of spectral properties of the operator coefficients. This can be done in the framework of evolution semigroups and sums of commuting operators in Section 2. Section 3 will be devoted to the critical case in which a fundamental technique of decomposition is presented. In Section 4 we will present another, but traditional, approach to periodic solutions of abstract functional differential equations. The remainder of the chapter will be devoted to several extensions of these methods to discrete systems and nonlinear equations.

3.1

3.1.1

Evolution Semigroups and Almost Periodic Solutions of Periodic Equations An Example

We begin this section with an example which is a trivial case where the system has an exponential dichotomy. Let us consider the equation 61

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dx = Ax + f (t), t ∈ R, x ∈ Rn , dt

(3.1)

where A is an n × n-matrix and f is a 1-periodic continuous function. We assume that the corresponding homogeneous equation of Eq.(3.1) has an exponential dichotomy. This is equivalent to the fact that σ(A) ∩ iR = , or S 1 ∩ σ(eA ) = (here S 1 denotes the unit circle in the complex plane). As is well known, in this case, for every almost periodic f , Eq.(3.1) has a unique bounded solution which can be represented in the form xf (t) =

Z

t −∞

etA P e−sA f (s)ds −

Z

+∞

etA (I − P )e−sA f (s)ds,

t

(3.2)

where the projection P is determined from the exponential dichotomy of the homogeneous equation (P is the spectral projection from Rn onto the invariant subspace corresponding to the part of eigenvalues of A with negative real parts). We now prove that this bounded solution is almost periodic. In fact, we will use the Bochner’s criterion. Let {tk }k∈N be any sequence of reals. We have to show that there is a subsequence {tkl } such that the sequence xf (tkl + ·) is convergent to a bounded uniformly continuous function uniformly with respect to t ∈ R. Note that, in this case, without loss of generality, we can assume that the projection P is commutative with the fundamental matrix etA . So, by assumption, P f (·) is almost periodic. By Bochner’s criterion, there is a subsequence of {tkl } such that P f (tkl + ·) is convergent uniformly. Let us consider the integral Z

t+tkl

e

(t+tkl −s)A

−∞

P f (s)ds =

Z

t

e(t−s)A P f (s + tkl )ds.

−∞

Now from the uniform convergence of f (tkl +·) and the absolute convergence of the integral in (3.2) follows the uniform convergence of the above integral. Similarly, we can prove the almost periodicity of the other integral in (3.2). Hence, xf is almost periodic. It is interesting to find the relationship between sp(f ) and sp(xf ). First, we assume that f (t) = eiλt , λ ∈ R, and n = 1 (hence A is 1 × 1-matrix with nonzero entry). Then we have (assuming that A = a < 0) Z

t −∞

ea(t−s) f (s)ds =

1 eiλt . iλ − a

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Therefore, sp(xf ) = {λ}. By using the Approximation Theorem, this proof can be extended to the general case of almost periodic f . In the general case we get the following inclusion: sp(xf ) ⊂ sp(f ) = σb (f ). The aim of our theory which will be presented here is to go futher in this direction. We will give conditions as sharp as possible for the existence of such almost periodic solutions as xf . We will show that the exponential dichotomy assumption on the homogeneous equation is too strong and redundant if the spectrum of f is a strict subset of R. Exercise 14. Show that under the above assumption we have σb (xf ) ⊂ σb (f ). Hint. Use Bohr transform. 3.1.2

Evolution Semigroups

Let us consider the following linear evolution equations dx = A(t)x, dt

(3.3)

and dx = A(t)x + f (t), (3.4) dt where x ∈ X, X is a complex Banach space, A(t) is a (unbounded) linear operator acting on X for every fixed t ∈ R such that A(t) = A(t + 1) for all t ∈ R , f : R → X is an almost periodic function. Under suitable conditions Eq.(3.3) is well-posed, i.e., one can associate with equation (3.3) an evolutionary process (U (t, s))t≥s which satisfies, among other things, the conditions in the following definition. Definition 3.1. A family of bounded linear operators (U (t, s))t≥s , (t, s ∈ R) from a Banach space X to itself is called 1-periodic strongly continuous evolutionary process if it is a strongly continuous evolutionary process and satisfies U (t + 1, s + 1) = U (t, s) for all t ≥ s. If it does not cause any danger of confusion, for the sake of simplicity, we shall often call 1-periodic strongly continuous evolutionary process (evolutionary) process. Note that the assumption that the period of the process is 1 is not a restriction. It is merely to shorten the notations.

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Once the well-posedness of the equations in question is assumed, instead of the equations with operator-coefficient A(t), we are in fact concerned with the evolutionary processes generated by these equations. In light of this, throughout the book we will deal with the asymptotic behavior of evolutionary processes as defined in Definition 3.1. Our main tool to study the asymptotic behavior of evolutionary processes is to use the notion of evolution semigroups associated with given evolutionary processes, which is defined in the following: Definition 3.2. The following formal semigroup associated with a given 1-periodic strongly continuous evolutionary process (U (t, s))t≥s (T h u)(t) := U (t, t − h)u(t − h),

∀t ∈ R, h ≥ 0,

(3.5)

where u is an element of some function space F, is called evolutionary semigroup associated with the process (U (t, s))t≥s on F. 3.1.3

The Finite Dimensional Case

We assume in this section that the evolutionary process (U (t, s))t≥s is generated by the ordinary differential equation dx = A(t)x, t ∈ R, x ∈ Cn , (3.6) dt where A(t) is 1-periodic continuous matrix function. Hence, by the Existence and Uniqueness Theorem, the fundamental matrix X(t) associated with (3.6) exists, i.e., the matrix satisfying the Cauchy problem ( dX dt = A(t)X, t ∈ R, (3.7) X(0) = I. Now setting U (t, s) = X(t)X −1 (s) we get the so-called Cauchy operators, or evolution operators associated with (3.6). We consider the evolution semigroup (T h )h≥0 associated with this evolutionary process (U (t, s))t≥s in the function space AP (X). First, note that, by the 1-periodicity of A(t) and the Existence and Uniqueness Theorem, U (t + 1, s + 1) = U (t, s), ∀t, s. Thus, it is seen that T h : AP (X) → AP (X), ∀h ≥ 0. Moreover, the strong continuity of the evolution semigroup can be proved easily by using classical rules of differentiation. Here we are interested in the infinitesimal generator L of this evolution semigroup. It is elementary

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to compute the generator by using the rules of differentation that is the following form: L = −d/dt + A(t), i.e., g ∈ D(L) ⊂ AP (X) if and only if g is differentiable and −g 0 (·) + A(·)g(·) ∈ AP (X).

This information suggests that the inhomogeneous equation dx = A(t)x + f (t), f ∈ AP (X) (3.8) dt has an almost periodic solution x(·) if and only if Lx = −f . This is very useful because if we can find conditions so that the operator L is invertible, i.e., 0 6∈ σ(L), then the inhomogeneous equation has a unique almost periodic solution. In turn, this condition can be found using the spectral inclusion of strongly continuous semigroups, i.e., 1 6∈ σ(T 1 ). This section will discuss the question as how to use the above ideas for the general infinite dimensional case. Exercise 15. Prove the formula for the generator of the evolution semigroup in the finite dimensional case. 3.1.4

The Infinite Demensional Case

In the infinite dimensional case many difficulties arise. First, the formula for the generator turns out to be more complicated. This is due to the fact that the Existence and Uniqueness Theorem for classical solutions does not applied to inhomogeneous equations. Hence, the classical rules of differentiation do not applied. Instead of the notion of classical solutions one introduces the one of mild solutions which saves several classical results in this case. Below we are mainly concerned with the following inhomogeneous equation Z t x(t) = U (t, s)x(s) + U (t, ξ)f (ξ)dξ, ∀t ≥ s (3.9) s

associated with a given strongly continuous 1-periodic evolutionary process (U (t, s))t≥s . A continuous solution u(t) of Eq.(3.9) on an interval J will be called mild solution to Eq.(3.4) on J. If we do not mention the interval J for a mild solution, we mean that the mild solution is defined on the whole real line.

The following lemma will be the key tool to study spectral criteria for almost periodicity in this section which relates the generator of the evolution semigroup (3.5) with the operator defined by Eq.(3.9).

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Lemma 3.1. Let (U (t, s))t≥s be a 1-periodic strongly continuous evolutionary process. Then its associated evolutionary semigroup (T h )h≥0 is strongly continuous in AP (X). Moreover, the infinitesimal generator of (T h )h≥0 is the operator L defined as follows: u ∈ D(L) and Lu = −f if and only if u, f ∈ AP (X) and u is the solution to Eq.(3.9). Proof. Let v ∈ AP (X). First we can see that T h acts on AP (X). To this end, we will prove the following assertion: Let Q(t) ∈ L(X) be a family of bounded linear operators which is periodic in t and strongly continuous, i.e., Q(t)x is continuous in t for every given x ∈ X. Then if f (·) ∈ AP (X), Q(·)f (·) ∈ AP (X). The fact that supt kQ(tk < ∞ follows from the Uniform Boundedness Principle. By the Approximation Theorem of almost periodic functions we can choose sequences of trigonometric polynomials fn (t) which converges uniformly to f (t) on the real line. For every n ∈ N, it is obvious that Q(·)fn (·) ∈ AP (X). Hence sup kQ(t)fn (t) − Q(t)f (t)k ≤ sup kQ(t)k sup kfn (t) − f (t)k t

t

t

implies the assertion. We continue our proof of Lemma 3.1. By definition we have to prove that lim sup kU (t, t − h)v(t − h) − v(t)k = 0.

h→0+

(3.10)

t

Since v ∈ AP (X) the range of v(·) which we denote by K (consisting of x ∈ X such that x = v(t) for some real t) is a relatively compact subset of X. Hence the map (t, s, x) 7→ U (t, s)x is uniformly continuous in the set {1 ≥ t ≥ s ≥ −1, x ∈ K}. Now let ε be any positive real. In view of the uniform continuity of the map (t, s, x) 7→ U (t, s)x in the above-mentioned set, there is a positive real δ = δ(ε) such that kU (t − [t], t − [t] − h)x − xk < ε

(3.11)

lim sup kU (t, t − h)v(t − h) − v(t − h)k = 0.

(3.12)

for all 0 < h < δ < 1 and x ∈ K, where [t] denotes the integer n such that n ≤ t < n + 1. Since (U (t, s))t≥s is 1-periodic from (3.11) this yields h→0+

t

Now we have lim sup kU (t, t − h)v(t − h) − v(t)k

h→0+

t

≤ lim sup kU (t, t − h)v(t − h) − v(t − h)k h→0+

t

+ lim+ sup kv(t − h) − v(t)k. h→0

t

(3.13)

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Since v is uniformly continuous this estimate and (3.13) imply (3.10), i.e., the evolutionary semigroup (T h )h≥0 is strongly continuous in AP (X). For the proof of the remainder we can do as in Section 2 of the previous chapter. To present another proof which works also for semilinear equations, let us consider the affine semigroup (Tfh )h≥0 associated with the inhomogeneous equation (3.9) for f ∈ AP (X), defined as follows: Z h Z h h h h−ξ h Tf v = T v + T f dξ = T v + T ξ f dξ, (3.14) 0

0

where v ∈ AP (X), h ≥ 0. It is easily checked that for all v ∈ AP (X) we have [Tfh v](t) = Uf (t, t − h)v(t − h), t ∈ R, h ≥ 0, where Uf (t, s) is the evolutionary operator defined by the integral equation (3.9). In other words, the assertion that g, f ∈ AP (X) and g is a solution of (3.9), is equivalent to Tfh g = g (∀h ≥ 0) . From (3.14) this is equivalent to Z h h h T ξ f dξ, ∀h ≥ 0 , g = Tf g = T g + 0

T hg − g =

Z

h

0

T ξ Agdξ = −

Z

0

h

T ξ f dξ ∀h ≥ 0.

(3.15)

From the general theory of linear operator semigroups (see Theorem 1.4) this is equivalent to the assertion Ag = −f . Remark 3.1. It may be noted that in the proof of Lemma 3.1 the precompactness of u and f are essiential. Hence, in the same way, we can show the strong continuity of the evolution semigroup (T h )h≥0 in C0 (R, X). Finally, combining this remark and Lemma 3.1 we get immediately the following corollary. Corollary 3.1. Let (U (t, s))t≥s be a 1-periodic strongly continuous process. Then its associated evolutionary semigroup (T h )h≥0 is a C0 -semigroup in AAP (X) := AP (X) ⊕ C0 (R, X). One of the interesting applications of Corollary 3.1 is the following. Corollary 3.2. Let (U (t, s)t≥s be a 1-periodic strongly continuous evolutionary process. Moreover, let u, f ∈ AAP (X) such that u is a solution of Eq.(3.9). Then the almost periodic component uap of u satisfies Eq.(3.9) with f := fap , where fap is the corresponding almost periodic component of f.

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Proof. The evolution semigroup (T h )h≥0 leaves the subspaces AP (X) and C0 (R, X) invariant. Let us denote by Pap , P0 the projections on these function spaces, respectively. Then since u is a solution to Eq.(3.9), by Lemma 3.1, lim+

h→0

T hu − u = −f. h

Hence, Pap lim

h→0+

T h Pap u − Pap u T hu − u = lim = −Pap f. h h h→0+

This, by Lemma 3.1, shows that Pap u := uap is a solution of Eq.(3.9) with f := Pap f := fap . 3.1.5

Almost Periodic Solutions and Applications

3.1.5.1 Invariant functions spaces of evolution semigroups Below we shall consider the evolutionary semigroup (T h )h≥0 in some special invariant subspaces M of AP (X). Definition 3.3. The subspace M of AP (X) is said to satisfy condition H if the following conditions are satisfied: (1) M is a closed subspace of AP (X), (2) There exists λ ∈ R such that M contains all functions of the form eiλ· x, x ∈ X, (3) If C(t) is a strongly continuous 1-periodic operator valued function and f ∈ M, then C(·)f (·) ∈ M, (4) M is invariant under the group of translations. In the sequel we will be mainly concerned with the following concrete examples of subspaces of AP (X) which satisfy condition H: Example 3.1. Let us denote by P(1) the subspace of AP (X) consisting of all 1-periodic functions. It is clear that P(1) satisfies condition H. Example 3.2. Let (U (t, s))t≥s be a strongly continuous 1-periodic evolutionary process. Hereafter, for every given f ∈ AP (X), we shall denote by M(f ) the subspace of AP (X) consisting of all almost periodic functions u such that sp(u) ⊂ {λ + 2πn, n ∈ Z, λ ∈ sp(f )}. Then M(f ) satisfies condition H.

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In fact, obviously, it is a closed subspace of AP (X), and moreover it satisfies conditions ii), iv) of the definition. We now check that condition iii) is also satisfied by proving the following lemma: Lemma 3.2. Let Q(t) be a 1-periodic operator valued function such that the map (t, x) 7→ Q(t)x is continuous. Then for every u(·) ∈ AP (X), the following spectral estimate holds true: sp(Q(·)u(·)) ⊂ Λ,

(3.16)

where Λ := {λ + 2kπ, λ ∈ sp(u), k ∈ Z}. Proof. Using the Approximation Theorem of almost periodic functions we can choose a sequence of trignometric polynomials N (m)

u

(m)

(t) =

X

eiλk,m t ak,m ,

k=1

ak,m ∈ X

such that λk,m ∈ σb (u) (:= Bohr spectrum of u), limm→∞ u(m) (t) = u(t) uniformly in t ∈ R. The lemma is proved if we have shown that sp(Q(·)u(m) (·)) ⊂ Λ.

(3.17)

In turn, to this end, it suffices to show that sp(Q(·)eiλk,m · ak,m ) ⊂ Λ.

(3.18)

In fact, since Q(·)ak,m is 1-periodic in t, there is a sequence of trignometric polynomials N (n)

Pn (t) =

X

k=−N (n)

ei2πkt pk,n , pk,n ∈ X

converging to Q(·)ak,m uniformly as n tends to ∞. Obviously, sp(eiλk,m · Pn (·)) ⊂ Λ.

(3.19)

Hence, sp(eiλk,m · Q(·)ak,m ) ⊂ Λ.

An important class of invariant subspaces is that of subspaces satisfying condition H. Proposition 3.1. Every subspace of AP (X) satisfying condition H is invariant under the evolution semigroup (T h )h≥0 associated with a given 1periodic strongly continuous evolutionary process on X.

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Proof.

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The proof is an easy exercise which is left to the reader.

The following corollary will be the key tool to study the unique solvability of the inhomogeneous equation (3.9) in various subspaces M of AP (X) satisfying condition H. Corollary 3.3. Let M satisfy condition H. Then, if 1 ∈ ρ(T 1 |M ), the inhomogeneous equation (3.9) has a unique solution in M for every f ∈ M. Proof. Under the assumption, the evolutionary semigroup (T h )h≥0 leaves M invariant. The generator A of (T h |M )h≥0 can be defined as the part of L in M. Thus, the corollary is an immediate consequence of Lemma 3.1 and the spectral inclusion eσ(A) ⊂ σ(T 1 |M ). 3.1.5.2 Monodromy operators In view of Corollary 3.3 the problem of finding conditions for the existence of a unique almost periodic mild solution to Eq.(3.4) is now reduced to that of finding conditions for 1 6∈ σ(T 1 |M ). To this end, we now analyse the spectrum of T 1 |M . By definition T 1 v(t) := U (t, t−1)v(t−1), so it is the composition of a translation and a multiplication operator. In the theory of ordinary differential equations the operator U (t, t − 1) is well studied and called monodromy operator (more precisely, the operator U (1, 0)). Exercise 16. In the finite dimensional case, i.e., the process (U (t, s))t≥s is the Cauchy operators of an ordinary differential equation, using the 1periodicity and the Existence and Uniqueness Theorem show that U (t, t − 1) = U (t − 1, 0)U (1, 0)U −1 (t − 1, 0) ∀t ∈ R. Proof.

Let U (t, 0) be the solution of the Cauchy problem ( dU (t) dt = A(t)U (t), U (0) = I.

(3.20)

(3.21)

Also, the operator U1 (t) = U (t + 1, 0)U −1 (1, 0) is another solution to this equation due to its 1-periodicity. Using the Existence and Uniqueness Theorem for ODE we see that U (t+1, 0)U −1 (1, 0) = U (t, 0). Now (3.20) follows from this. Hence, the spectrum of U (t, t − 1) is the same as that of U (1, 0). In the infinite dimensional case, in general the process is not invertible, so this property does not hold. However, the spectral properties of U (1, 0) and U (t, t − 1) are almost the same.

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First we collect some results which we shall need in the book. Recall that for a given 1-periodic evolutionary process (U (t, s))t≥s the following operator P (t) := U (t, t − 1), t ∈ R

(3.22)

is called monodromy operator (or sometime, period map, Poincar´e map). Thus we have a family of monodromy operators. Throughout the paper we will denote P := P (0). The nonzero eigenvalues of P (t) are called characteristic multipliers. An important property of monodromy operators is stated in the following lemma. Lemma 3.3. Under the notation as above the following assertions hold: (1) P (t + 1) = P (t) for all t; characteristic multipliers are independent of time, i.e. the nonzero eigenvalues of P (t) coincide with those of P , (2) σ(P (t))\{0} = σ(P )\{0}, i.e., it is independent of t, (3) If λ ∈ ρ(P ), then the resolvent R(λ, P (t)) is strongly continuous. Proof. The periodicity of P (t) is obvious. In view of this property we will consider only the case 0 ≤ t ≤ 1. Suppose that µ 6= 0, P x = µx 6= 0, and let y = U (t, 0)x, so U (1, t)y = µy 6= 0, y 6= 0 and P (t)y = µy. By the periodicity this shows the first assertion. Let λ 6= 0 belong to ρ(P ). We consider the equation λx − P (t)x = y,

(3.23)

where y ∈ X is given. If x is a solution to Eq.(3.23), then λx = y +w, where w = U (t, 0)(λ − P )−1 U (1, t)y. Conversely, defining x by this equation, it follows that (λ − P (t))x = y so ρ(P (t)) ⊃ ρ(P )\{0} . The second assertion follows by the periodicity. Finally, the above formula involving x proves the third assertion. By this lemma, once we are interested in the spectrum of P (t) rather than the operators P (t) for different t, we see that they are almost the same. So, by monodromy operator we may understand the operator P := P (0) for convenience if this does not cause any confusion. Let M be a subspace of AP (X) invariant under the evolution semigroup (T h )h≥0 associated with the given 1-periodic evolutionary process (U (t, s))t≥s in AP (X). Below we will use the following notation PˆM v(t) := P (t)v(t), ∀t ∈ R, v ∈ M.

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If M = AP (X) we will denote PˆM = Pˆ . In the sequel we need the following lemma: Lemma 3.4. Let (U (t, s))t≥s be a 1-periodic strongly continuous evolutionary process and M be an invariant subspace of the evolution semigroup (T h )h≥0 associated with it in AP (X). Then for all invariant subspaces M satisfying condition H, σ(PˆM )\{0} = σ(P )\{0}. Proof. For u, v ∈ M , consider the equation (λ− PˆM )u = v . It is equivalent to the equation (λ − P (t))u(t) = v(t), t ∈ R. If λ ∈ ρ(PˆM )\{0}, for every v the first equation has a unique solution u, and kuk ≤ kR(λ, PˆM )kkvk. Take a function v ∈ M of the form v(t) = yeiµt , for some µ ∈ R ; the existence of such a µ is guaranteed by the axioms of condition H. Then the solution u satisfies kuk ≤ kR(λ, PˆM )kkyk. Hence, for every y ∈ X the solution of the equation (λ − P (0))u(0) = y has a unique solution u(0) such that ku(0)k ≤ sup ku(t)k ≤ kR(λ, PˆM )k sup kv(t)k ≤ kR(λ, PˆM )kkyk. t

t

This implies that λ ∈ ρ(P )\{0} and kR(λ, P (t))k ≤ kR(λ, PˆM )k . Conversely, suppose that λ ∈ ρ(P )\{0}. By Lemma 3.3 for every v the second equation has a unique solution u(t) = R(λ, P (t))v(t) and the map taking t into R(λ, P (t)) is strongly continuous. By definition of condition H, the function taking t into (λ − P (t))−1 v(t) belongs to M. Since R(λ, P (t)) is a strongly continuous, 1-periodic function, by the uniform boundedness principle it holds that r := sup{kR(λ, P (t))k : t ∈ R} < ∞. This means that ku(t)k ≤ rkv(t)k ≤ r supt kv(t)k, or kuk ≤ rkvk. Hence λ ∈ ρ(PˆM ), and kR(λ, PˆM )k ≤ r . Remark 3.2. By similar argument as above we can show that if P is the operator of multiplication by P (t) in C0 (R, X) then σ(P)\{0} = σ(P )\{0}. As a consequence of this we have Corollary 3.4. Let (U (t, s))t≥s be a strongly continuous 1-periodic evolutionary process with monodromy operator P and let σ(P ) be disjoint from the unit circle. Then, the process (U (t, s))t≥s has an exponential dichotomy. Proof. By the above remark we have that σ(P) is disjoint from the unit circle as well. Let us denote the translation group in C0 (R, X) by (S(t))t∈R . Then, T 1 v = PS(−1)v, h ≥ 0, v ∈ C0 (R, X).

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Note that S(−1) and P are bounded linear operators that commute with each other. Therefore, σ(T 1 ) = σ(PS(−1)) ⊂ σ(P) · σ(S(−1)). From the Spectral Radius Theorem it is easy to see that σ(S(−1)) is contained in (actually coincides with) the unit circle. Hence, σ(T 1 ) is disjoint from the unit circle. By Theorem 2.2, the process (U (t, s))t≥s has an exponential dichotomy. 3.1.5.3 Unique solvability of the inhomogeneous equations in P(1) We now illustrate Corollary 3.2 in some concrete situations. First we will consider the unique solvability of Eq.(3.9) in P(1). Proposition 3.2. Let (U (t, s))t≥s be 1-periodic strongly continuous. Then the following assertions are equivalent: (i) 1 ∈ ρ(P ), (ii) Eq.(3.9) is uniquely solvable in P(1) for a given f ∈ P(1). Proof. Suppose that i) holds true. Then we show that ii) holds by applying Corollary 3.2. To this end, we show that σ(T 1 |P(1) )\{0} ⊂ σ(P )\{0} . To see this, we note that T 1 |P(1) = PˆP(1) . In view of Lemma 3.4 1 ∈ ρ(T 1 |P(1) ). By Example 3.1 and Corollary 3.2 ii) holds also true. Conversely, we suppose that Eq.(3.9) is uniquely solvable in P(1). We now show that 1 ∈ ρ(P ). For every x ∈ X put f (t) = U (t, 0)g(t)x for t ∈ [0, 1], where g(t) is any continuous function of t such that g(0) = g(1) = 0, and Z 1

g(t)dt = 1.

0

Thus f (t) can be continued to a 1-periodic function on the real line which we denote also by f (t) for short. Put Sx = [L−1 (−f )](0) . Obviously, S is a bounded operator. We have Z 1 −1 −1 [L (−f )](1) = U (1, 0)[L (−f )](0) + U (1, ξ)U (ξ, 0)g(ξ)xdξ 0

Sx = P Sx + P x.

Thus (I − P )(Sx + x) = P x + x − P x = x. So, I − P is surjective. From the uniqueness of solvability of (3.9) we get easily the injectiveness of I − P . In other words, 1 ∈ ρ(P ).

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3.1.5.4 Unique solvability in AP (X) and exponential dichotomy This subsection will be devoted to the unique solvability of Eq.(3.9) in AP (X) and its applications to the study of exponential dichotomy. Let us begin with the following lemma which is a consequence of Proposition 3.2. Lemma 3.5. Let (U (t, s))t≥s be 1-periodic strongly continuous. Then the following assertions are equivalent: (i) S 1 ∩ σ(P ) = . (ii) For every given µ ∈ R, f ∈ P(1) the following equation has a unique solution in AP (X) Z t U (t, ξ)eiµξ f (ξ)dξ, ∀t ≥ s. (3.24) x(t) = U (t, s)x(s) + s

Proof.

Suppose that i) holds, i.e S 1 ∩ σ(P ) = . Then, since T 1 = S(−1) · Pˆ = Pˆ · S(−1)

in view of the commutativeness of two operators Pˆ and S(−1) we have σ(T 1 ) ⊂ σ(S(−1)).σ(Pˆ ). It may be noted that σ(S(−1)) = S 1 . Thus σ(T 1 ) ⊂ {eiµ λ, µ ∈ R, λ ∈ σ(Pˆ )}. Hence, in view of Lemma 3.4 σ(T 1 ) ∩ S 1 = . Let us consider the process (V (t, s))t≥s defined by V (t, s)x := e−iµ(t−s) U (t, s)x for all t ≥ s, x ∈ X. Let Q(t) denote its monodromy operator, i.e. Q(t) = e−iµ V (t, t−1) and (Tµh )h≥0 denote the evolution semigroup associated with the evolutionary process (V (t, s))t≥s . Then by the same argument as above we can show that since σ(Tµh ) = e−iµ σ(T h ), σ(Tµh ) ∩ S 1 = . By Lemma 3.1 and Corollary 3.2, the following equation Z t y(t) = V (t, s)y(s) + V (t, ξ)f (ξ)dξ, ∀t ≥ s s

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has a unique almost periodic solution y(·) . Let x(t) := eiµt y(t) . Then Z t x(t) = eiµt y(t) = U (t, s)eiµs y(s) + U (t, ξ)eiµξ f (ξ)dξ s Z t iµξ U (t, ξ)e f (ξ)dξ ∀t ≥ s. = U (t, s)x(s) + s

Thus x(·) is an almost periodic solution of Eq.(3.24). The uniqueness of x(·) follows from that of the solution y(·) . We now prove the converse. Let y(t) be the unique almost periodic solution to the equation Z t U (t, ξ)eiµξ f (ξ)dξ, ∀t ≥ s. (3.25) y(t) = U (t, s)y(s) + s

Then x(t) := e tion

−iµt

y(t) must be the unique solution to the following equa-

x(t) = e−iµ(t−s) U (t, s)x(s) +

Z

t s

e−iµ(t−ξ) U (t, ξ)f (ξ)dξ), ∀t ≥ s. (3.26)

And vice versa. We show that x(t) should be periodic. In fact, it is easily seen that x(1 + ·) is also an almost periodic solution to Eq.(3.25). From the uniqueness of y(·) (and then that of x(·)) we have x(t + 1) = x(t), ∀t. By Proposition 3.2 this yields that 1 ∈ ρ(Q(0)), or in other words, eiµ ∈ ρ(P ). From the arbitrary nature of µ, S 1 ∩ σ(P ) = . Theorem 3.1. Let (U (t, s))t≥s be given 1-periodic strongly continuous evolutionary process. Then the following assertions are equivalent: (i) The process (U (t, s))t≥s has an exponential dichotomy; (ii) For every given bounded and continuous f the inhomogeneous equation (3.9) has a unique bounded solution; (iii) The spectrum of the monodromy operator P does not intersect the unit circle; (iv) For every given f ∈ AP (X) the inhomogeneous equation (3.9) is uniquely solvable in the function space AP (X) . Proof. The equivalence of i) and ii) has been established in Theorem 2.1 and the remarks that follow. Now we show the equivalence between i), ii) and iii). The fact that iii) implies i) is the content of Corollary 3.4. Now we prove that i) implies iii). Let the process have an exponential dichotomy. We now show that the spectrum of the monodromy operator P does not intersect the unit circle. We will follow the argument of Lemma 3.5. Since

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i) is equivalent to ii), for every fixed real number µ and f ∈ P(1), there exists a unique bounded solution y to (3.25). By the argument of Lemma 3.5, the function x(t) := e−iµt y(t) must be the unique bounded solution to (3.26) and that should be 1-periodic. Therefore, actually the solution y is almost periodic. By Lemma 3.5, S 1 ∩ σ(P ) = ∅. So i) is equivalent to iii). It remains to show the equivalence between iii) and iv). We first show the implication iii) ⇒ iv). By Lemma 3.4 the operator Pˆ of multiplication by P (t) on AP (X) has the property that σ(Pˆ ) ∩ S 1 = ∅. Therefore, since Pˆ commutes with translation operator S(t) the evolution semigroup T h on AP (X) has the property that σ(T 1 ) = σ(Pˆ S(−1)) ⊂ σ(Pˆ ) · σ(S(−1)). Since σ(S(−1)) = S 1 we have that σ(T 1 ) ∩ S 1 = ∅. By the Spectral Inclusion Theorem for C0 -semigroup and Lemma 3.1 we have iv). Conversely, let iv) hold. Then by Lemma 3.5 we have iii). 3.1.5.5 Unique solvability of the inhomogeneous equations in M(f ) Now let us return to the more general case where the spectrum of the monodromy operator may intersect the unit circle. Theorem 3.2. Let (U (t, s))t≥s be a 1-periodic strongly continuous evolutionary process. Moreover, let f ∈ AP (X) such that σ(P ) ∩ iλ {e , λ ∈ sp(f )} = . Then the inhomogeneous equation (3.9) has an almost periodic solution which is unique in M(f ) . Proof. From Example 3.2 it follows that the function space M(f ) satisfies condition H. Since (S(t))t∈R is an isometric C0 -group, by the weak spectral mapping theorem for isometric groups (see e.g. [Nagel (73)]) we have σ(S(1)|M(f ) ) = eσ(D|M(f ) ) , where D|M(f ) is the generator of (S(t)|M(f ) )t≥0 . From the general spectral theory of bounded functions we have σ(D|M(f ) ) = iΛ, where Λ = {λ + 2πk, λ ∈ sp(f ), k ∈ Z}. Hence, since eσ(D|M(f ) ) = eiΛ ⊂ eisp(f ) ⊂ eiΛ ,

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we have σ(S(1)|M(f ) ) = eσ(D|M(f ) ) = eisp(f ) . Thus, the condition σ(P ) ∩ eisp(f ) = is equivalent to the following 1 6∈ σ(P ).σ(S(−1)|M(f ) ). In view of the inclusion σ(T 1 |M(f ) )\{0} ⊂ σ(PˆM(f ) ).σ(S(−1)|M(f ) )\{0} ⊂ σ(P ).σ(S(−1)|M(f ) )\{0}

which follows from the commutativeness of the operator PˆM(f ) with the operator S(−1)|M(f ) , the above inclusion implies that 1 6∈ σ(T 1 |M(f ) ). Now the assertion of the theorem follows from Corollary 3.2.

The inverse of the above theorem is the following: Theorem 3.3. Let (U (t, s))t≥s be a 1-periodic strongly continuous evolutionary process. Moreover, let the inhomogeneous equation (3.9) have a unique almost periodic solution in M(f ). Then σ(P ) ∩ {eiλ , λ ∈ sp(f )} = . Proof. Let µ ∈ sp(f ). Consider the function fλ (t) := eiµt f (t), where f (t) ∈ P(1). Then there is a unique solution xµ (·) ∈ AP (X) of the equation (3.24), or equivalently, a unique solution y(t) = e−iµt x(t) to the equation (3.26). Since, the equation (3.26) is periodic in t, if there is a solution y(·), then y(1+·) is also a solution. From the uniqueness it follows that y(1+·) = y(·). This means that y(·) is 1-periodic. Hence, for every f ∈ P(1) Eq.(3.26) has a unique 1-periodic solution y(·). By Proposition 3.2 0 6∈ σ(Q(0)) which is equivalent to eiµ 6∈ σ(P ). Hence, {eiµ , µ ∈ sp(f )} ∩ σ(P ) = . 3.1.5.6 Unique solvability of nonlinearly perturbed equations Let us consider the semilinear equation Z t x(t) = U (t, s)x(s) + U (t, ξ)g(ξ, x(ξ))dξ. s

(3.27)

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We shall be interested in the unique solvability of (3.27) for a larger class of the forcing term g. We shall show that the generator of evolutionary semigroup is still useful in studying the perturbation theory in the critical case in which the spectrum of the monodromy operator P may intersect the unit circle. We suppose that g(t, x) is Lipschitz continuous with coefficient k and the Nemystky operator F defined by (F v)(t) = g(t, v(t)), ∀t ∈ R acts in M. Below we can assume that M is any closed subspace of the space of all bounded continuous functions BC(R, X). We consider the operator L in BC(R, X). If (U (t, s))t≥s is strongly continuous, then L is a single-valued operator from D(L) ⊂ BC(R, X) to BC(R, X). Lemma 3.6. Let M be any closed subspace of BC(R, X), (U (t, s))t≥s be strongly continuous and Eq.(3.9) be uniquely solvable in M. Then for sufficiently small k , Eq.(3.27) is also uniquely solvable in this space. Proof. First, we observe that under the assumptions of the lemma we can define a single-valued operator L acting in M as follows: u ∈ D(L) if and only if there is a function f ∈ M such that Eq.(3.9) holds. From the strong continuity of the evolutionary process (U (t, s))t≥s one can easily see that there is at most one function f such that Eq.(3.9) holds. This means L is single-valued. Moreover, one can see that L is closed. Now we consider the Banach space [D(L)] with graph norm, i.e. |v| = kvk + kLvk. By assumption it is seen that L is an isomorphism from [D(L)] onto M. In view of the Lipschitz Inverse Mapping Theorem for Lischitz mappings for sufficiently small k the operator L − F is invertible. Hence there is a unique u ∈ M such that Lu − F u = 0. From the definition of operator L we see that u is a unique solution to Eq.(3.27). Corollary 3.5. Let M be any closed subspace of AP (X), (U (t, s))t≥s be 1-periodic strongly continuous evolutionary process and for every f ∈ M the inhomogeneous equation (3.9) be uniquely solvable in M . Moreover let the Nemytsky operator F induced by the nonlinear function g in Eq.(3.27) act on M . Then for sufficiently small k , the semilinear equation (3.27) is uniquely solvable in M . Proof.

The corollary is an immediate consequence of Lemma 3.6.

3.1.5.7 Example 1 In this example we shall consider the abstract form of parabolic partial differential equations (see e.g. [Henry (46)]) and apply the results obtained

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above to study the existence of almost periodic solutions to these equations. It may be noted that a necessary condition for the existence of Floquet representation is that the process under consideration is invertible. It is known for the bounded case (see e.g. Chap. V, Theorem 1.2 in [Daleckii and Krein (24)]) that if the spectrum of the monodromy operator does not circle the origin (of course, it should not contain the origin), then the evolution operators admit Floquet representation. In the example below, in general, Floquet representation does not exist. For instance, if the sectorial operator A has compact resolvent, then monodromy operator is compact (see [Henry (46)] for more details). Thus, if dimX = ∞, then monodromy operators cannot be invertible. However, the above results can apply. Let A be sectorial operator in a Banach space X, and the mapping taking t into B(t) ∈ L(Xα , X) be H¨ older continuous and 1-periodic. Then there is a 1-periodic evolutionary process (U (t, s))t≥s associated with the equation du = (−A + B(t))u. (3.28) dt We have the following: Claim 1 For any x0 ∈ X and τ there exists a unique (strong) solution x(t) := x(t; τ, x0 ) of Eq.(3.28) on [τ, +∞) such that x(τ ) = x0 . Moreover, if we write x(t; τ, x0 ) := T (t, τ )x0 , ∀t ≥ τ , then (T (t, τ ))t≥τ is a strongly continuous 1-periodic evolutionary process. In addition, if A has compact resolvent, then the monodromy operator P (t) is compact. Proof. This claim is an immediate consequence of Theorem 7.1.3, pp. 190-191 in [Henry (46)]. In fact, it is clear that (T (t, τ ))t≥τ is strongly continuous and 1-periodic. The last assertion is contained in Lemma 7.2.2, p. 197 in [Henry (46)]). Thus, in view of the above claim if dimX = ∞, then Floquet representation does not exist for the process. This means the problem cannot reduced to the autonomous and bounded case. To apply our results, let the function f taking t into f (t) ∈ X be almost periodic and the spectrum of the monodromy operator of the process (U (t, s))t≥s be separated from the set eisp(f ) . Then the following inhomogeneous equation du = (−A + B(t))u + f (t) dt has a unique almost periodic solution u such that sp(u) ⊂ {λ + 2πk, k ∈ Z, λ ∈ sp(f )}.

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We now show Claim 2 Let the conditions of Claim 1 be satisfied except for the compactness of the resolvent of A . Then dx = (−A + B(t))x (3.29) dt has an exponential dichotomy if and only if the spectrum of the monodromy operator does not intersect the unit circle. Moreover, if A has compact resolvent, it has an exponential dichotomy if and only if all multipliers have modulus different from one. In particular, it is asymptotically stable if and only if all characteristic multipliers have modulus less than one. Proof. The operator T (t, s), t > s is compact if A has compact resolvent (see e.g. p. 196 in [Henry (46)]). The claim is an immediate consequence of Theorem 3.1. 3.1.5.8 Example 2 We examine in this example how the condition of Theorem 3.2 cannot be dropped. In fact we consider the simplest case with A = 0 dx = f (t), x ∈ R, (3.30) dt where f is continuous and 1-periodic. Obviously, σ(eA ) = {1} = ei sp(f ) . Rt We assume further that the integral 0 f (ξ)dξ is bounded. Then every solution to Eq.(3.30) can be extended to a periodic solution defined on the whole line of the form Z t

x(t) = c +

0

f (ξ)dξ, t ∈ R.

Thus the uniqueness of a periodic solution to Eq.(3.30) does not hold.

Now let us consider the same Eq.(3.30) but with 1-anti-periodic f , i.e., f (t + 1) = f (t), ∀t ∈ R. Clearly, ei sp(f ) = {−1} ∩ σ(eA ) = . Hence the conditions of Theorem 3.2 are satisfied. Recall that in this theorem we claim that the uniqueness of the almost periodic solutions is among the class of almost periodic functions g with ei sp(g) ⊂ ei sp(f ) . Now let us have a look at our example. Every solution to Eq.(3.30) is a sum of the unique 1-anti-periodic solution, which existence is guaranteed by Theorem 3.2, and a solution to the corresponding homogeneous equation, i.e., in this case a constant function. Hence, Eq.(3.30) has infinitely many almost periodic solutions.

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81

Evolution Semigroups and Sums of Commuting Operators

Let X be a given complex Banach space and M be a translation invariant subspace of the space of X-valued bounded uniformly continuous functions on the real line (that is denoted by BU C(R, X)). The problem we consider in this section is to find conditions for M to be admissible with respect to differential equations of the form du = Au + f (t), dt

(3.31)

where A is an (unbounded) linear operator with nonempty resolvent set on the Banach space X. By a tradition, by admissibility here we mean that for every f ∈ M Eq.(3.31) has a unique solution (in a suitable sense) which belongs to M as well. The main condition obtained in this section is of the form σ(A) ∩ isp(f ) = ∅. We will show that the method of sums of commuting operators can be extended to larger classes of equations, including abstract functional differential equations. 3.2.1

Invariant Function Spaces

By (S(t))t∈R we denote the translation group on the function space BU C(R, X), i.e., S(t)v(s) := v(t + s), ∀t, s ∈ R, v ∈ BU C(R, X) with infinitesimal generator D := d/dt defined on D(D) := BU C 1 (R, X). Let M be a subspace of BU C(R, X), and let A be a linear operator on X. We will denote by AM the operator f ∈ M 7→ Af (·) with D(AM ) = {f ∈ M|∀t ∈ R, f (t) ∈ D(A), Af (·) ∈ M}. When M = BU C(R, X) we will use the notation A := AM . Throughout the paragraph we always assume that A is a given operator on X with ρ(A) 6= , (so it is closed). In this paragraph we will use the notion of translation-invariance of a function space, which we recall in the following definition, and additional conditions on it. Definition 3.4. A closed and translation invariant subspace M of the function space BU C(R, X), i.e., S(τ )M ⊂ M for all τ ∈ R, is said to satisfy

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(1) condition H1 if the following condition is fulfilled: ∀C ∈ L(X), ∀f ∈ M ⇒ Cf ∈ M, (2) condition H2 if the following condition is fulfilled: For every closed linear operator A, if f ∈ M such that f (t) ∈ D(A), ∀t, Af ∈ BU C(R, X), then Af ∈ M, (3) condition H3 if the following condition is fulfilled: For every bounded linear operator B ∈ L(BU C(R, X)) which commutes with the translation group (S(t))t∈R one has BM ⊂ M. Remark 3.3. As remarked in p.401 in [Vu and Schuler (103)], condition H3 is equivalent to the assertion that ∀B ∈ L(M, X) ∀f ∈ M ⇒ BS(·)f ∈ M. Obviously, conditions H2, H3 are stronger than condition H1. In the sequel, we will define the autonomousness of a functional operator via condition H3. Example 3.3. If Λ(X) = {f ∈ BU C(R, X) : sp(f ) ⊂ Λ} , where Λ is a given closed subset of R. Then Λ(X) is a translation invariant closed subspace of BU C(R, X). Moreover, it satisfies all conditions H1, H2, H3. In connection with the translation-invariant subspaces we need the following simple spectral properties. Lemma 3.7. (i) Let M satisfy condition H1. Then σ(AM ) ⊂ σ(A) = σ(A) and kR(λ, AM )k ≤ kR(λ, A)k = kR(λ, A)k, ∀λ ∈ ρ(A); (ii) Let M satisfy condition H3 and B be a bounded linear operator on BU C(R, X) which commutes with the translation group. Then σ(BM ) ⊂ σ(B) and kR(λ, BM )k ≤ kR(λ, B)k, ∀λ ∈ ρ(B). Proof. i) Let λ ∈ ρ(A). We show that λ ∈ ρ(AM ). In fact, as M satisfies condition H1, ∀f ∈ M, R(λ, A)f (·) := (λ − A)−1 f (·) ∈ M. Thus the function R(λ, A)f (·) is a solution to the equation (λ − AM )u = f . Moreover, since λ ∈ ρ(A) it is seen that the above equation has at most

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one solution. Hence λ ∈ ρ(AM ). Moreover, it is seen that kR(λ, AM )k ≤ kR(λ, A)k. Similarly, we can show that if λ ∈ ρ(A), then λ ∈ ρ(A) and kR(λ, A)k ≤ kR(λ, A)k. ii) The proof of the second assertion can be done in the same way. In the section, as a model of the translation - invariant subspaces, which satisfy all conditions H1, H2, H3 we can take the spectral spaces Λ(X) := {u ∈ BU C(R, X) : sp(u) ⊂ Λ}, where Λ is a given closed subset of the real line. 3.2.2

Differential Operator d/dt − A and Notions of Admissibility

We start the main subsection of this section by discussing various notions of admissibility and their inter-relations via the differential operator d/dt − A, or more precisely its closed extensions, for the following equation dx = Ax + f (t), x ∈ X, t ∈ R, dt

(3.32)

where A is a linear operator acting on X. We first recall that Definition 3.5. (1) An X-valued function u on R is said to be a solution on R to Eq.(3.32) for given linear operator A and f ∈ BU C(R, X) (or sometime, classical solution) if u ∈ BU C 1 (R, X), u(t) ∈ D(A), ∀t and u satisfies Eq.(3.32) for all t ∈ R. (2) Let A be the generator of a C0 semigroup of linear operators. An Xvalued continuous function u on R is said to be a mild solution on R to Eq.(3.32) for a given f ∈ BU C(R, X) if u satisfies Z t (t−s)A u(t) = e u(s) + e(t−r)A f (r)dr, ∀t ≥ s. s

Definition 3.6. (1) A closed translation invariant subspace M ⊂ BU C(R, X) is said to be admissible for Eq.(3.32) if for each f ∈ M0 := M ∩ BU C 1 (R, X) there is a unique solution u ∈ M0 of Eq.(3.32) and if fn ∈ M0 , n ∈ N, fn → 0 as n → ∞ in M0 imply un → 0 as n → ∞.

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(2) Let A be the generator of a C0 -semigroup. A translation - invariant closed subspace M of BU C(R, X) is said to be mildly admissible for Eq.(3.32) if for every f ∈ M there exists a unique mild solution xf ∈ M to Eq.(3.32). Exercise 17. By definition it is obvious that admissibility implies that if DM − AM is closable, one has 0 ∈ ρ(DM − AM ). Proof. Set B := DM − AM . Obviously, B is injective. In fact, if Bf = 0 then from the uniqueness assumption f = 0. By assumption, from the continuity of the Green operator G, defined as the unique extension of f ∈ M0 7→ u ∈ M0 it follows that there exists m > 0 such that kBxk ≥ kxk for all x ∈ D(B). We now show that B is injective. Indeed, it sufficies to show that if By = 0, then y = 0. By definition, there exist (yn , xn ) ∈ Γ(B) such that yn → y, Byn = xn → 0 = By. Hence, 0 = lim n → ∞kxn k = kByk ≥ mkyk, so y = 0. We show that B is surjective. Let f ∈ M. We show that Gf is in D(B and BGf = f . In fact, by definition of the extension G, there are fn ∈ M0 such that fn → f and Gfn → Gf . Obviously, BGfn = BGfn = fn . This completes the proof of the exrcise. We now discuss the relationship between the notions of admissibility, weak admissibility and mild admissibility if A is the generator of a C0 semigroup. To this end, we introduce the following operator LM which will be the key tool in our construction. Definition 3.7. Let M be a translation invariant closed subspace of BU C(R, X). We define the operator LM on M as follows: u ∈ D(LM ) if and only if u ∈ M and there is f ∈ M such that Z t u(t) = e(t−s)A u(s) + e(t−r)A f (r)dr, ∀t ≥ s (3.33) s

and in this case LM u := f . The following lemma will be needed in the sequel Lemma 3.8. Let A be the generator of a C0 -semigroup and M be a closed translation invariant subspace of AAP (X) which satisfies condition H1. Then DM − A M = L M .

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Proof.

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Let us consider the semigroup (T h )h≥0 T h v(t) := ehA v(t − h), v ∈ M, h ≥ 0.

By condition H1, clearly, (T h )h≥0 leaves M invariant. By Corollary 3.1, since M ⊂ AAP (X) this semigroup is strongly continuous which has −LM as its generator. On the other hand, since (T h )h≥0 is the composition of two commuting and strongly continuous semigroups, by p. 24 in [Nagel (73)] this generator is nothing but −DM + AM . Corollary 3.6. Let A be the generator of a C0 -semigroup and M be a translation invariant closed subspace of BU C(R, X). Then the notions of admissibility and mild admissibility of M for Eq.(3.32) are equivalent provided that M satisfies condition H1 and M ⊂ AAP (X). Proof. Since, by Exercise 17 the admissibility of M for Eq.(3.32) implies in particular that 0 ∈ ρ(DM − AM ), and by Lemma 3.8 A

DM − A M = D M − A M = L M the implication ”admissibility ⇒ mild admissibility” is clear. It remains only to show ”mild admissibility ⇒ admissibility”, i.e., if 0 ∈ ρ(LM ), then M is admissible with respect to Eq.(3.32). In fact, by assumption, for every f ∈ M there is a unique mild solution u := L−1 M f of Eq.(3.32). It can be seen that the function u(τ + ·) ∈ M is a mild solution of Eq.(3.32) with the forcing term f (τ + ·) for every fixed τ ∈ R. Hence, by the uniqueness, u(τ + ·) = L−1 M f (τ + ·). We can rewrite this fact as −1 S(τ )L−1 M f = LM S(τ )f, ∀f ∈ M, τ ∈ R.

From this and the boundedness of L−1 M, lim+

τ →0

S(τ )f − f S(τ )u − u = L−1 . M lim+ τ τ τ →0

Thus, the assumption that f ∈ M0 implies that the left hand side limit Thus, u = L−1 M f ∈ M0 . As is well known, since f is differentiable Rexists. t (t−ξ)A e f (ξ)dξ is differentiable (see Theorem, p. 84 in [Goldstein (36)]). s Thus, by definition of mild solutions, from the differentiability of u it follows that e(t−s)A u(s) is differentiable with respect to t ≥ s. Thus, u(s) ∈ D(A) for every s ∈ R. Finally, this shows that u(·) is a classical solution to Eq.(3.32) on R. Hence the admissibility of M for Eq.(3.32) is proved.

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3.2.3

Admissibility for Equations

Abstract

Ordinary Differential

In this subsection we shall demonstrate some advantages of using the operator d/dt − A as the sum of two commuting operators to study the admissibility theory for Eq.(3.32). In the sequel we shall need the following basic property of the translation group on Λ(X) which proof can be done in a standard manner. Lemma 3.9. Let Λ be a closed subset of the real line. Then σ(DΛ(X) ) = iΛ. Proof. First, we note that for every λ ∈ Λ, iλ ∈ σ(DΛ(X) ). In fact, Deiλ· x = iλeiλ· x. Now suppose that λ0 6∈ Λ. Then we shall show that iλ0 ∈ ρ(DΛ(X) ). To this end, we consider the following equation du = iλ0 u + g(t), g ∈ Λ(X). (3.34) dt Since isp(g) = σ(DMg ), where Mg is the closed subspace of BU C(R, X), spanned by all translations of g (see e.g., [Arendt and Batty (5)],[Vu (101)]), we get iλ0 6∈ σ(DMg ); and hence the above equation has a unique solution h ∈ Mg ⊂ Λ(X). If k is another solution to Eq.(3.34) in Λ(X), then h − k is a solution in Λ(X) to the homogeneous equation associated with Eq.(3.34). Thus, a computation via Carleman transform shows that sp(h − k) ⊂ {λ0 }. On the one hand, we get λ0 6∈ sp(h − k) because of sp(h − k) ⊂ Λ. Hence, sp(h − k) = , and then h − k = 0. In other words, Eq.(3.34) has a unique solution in Λ(X). This shows that the above equation has a unique solution in Λ(X) , i.e. iλ0 ∈ ρ(DΛ(X) ). Theorem 3.4. Let A be the generator of a C0 -semigroup (T (t))t≥0 of linear operators on X and let Λ(X) be mildly admissible for Eq.(3.32). Then, iΛ ∩ σ(A) = . Proof. Suppose that G is the operator which takes every g ∈ Λ(X) into the unique mild solution uf of Eq. (3.32). We show that G commutes with the translation group (S(t))t∈R . In fact, for every τ ∈ R we see that if f ∈ Λ(X), then [Gf ](τ + t) = T ((τ + t) − (τ + s))[Gf ](τ + s) Z τ +t + T (τ + t − ξ)f (ξ)dξ, ∀t ≥ s τ +s

= T (t − s)[Gf ](τ + s) +

Z

t

s

T (t − η)f (τ + η)dη,

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and hence, the function u(t) = [Gf ](τ + t) = [S(τ )Gf ](t) is the unique solution to Eq. (3.32) with the forcing term f (τ + t). That is S(τ )Gf = GS(τ )f , i.e., G commutes with the translation group (S(t))t∈R . Let λ ∈ Λ. We consider the function f (t) := aeiλt , t ∈ R with 0 6= a ∈ X. By the above remark on the commutativeness of G and (S(t))t∈R , we have S(τ )Gf − Gf dGf = lim τ →0 dt τ S(τ )f − f = lim G τ →0 τ S(τ )f − f = G lim τ →0 τ = iλGf. This shows that u(t) := [Gf ](t) = beiλt . Substituting this expression into the equation Z t u(t) = T (t − s)u(s) + T (t − s)f (s)ds, s

we have

b = R(h)b +

Z

h 0

R(h − ξ)adξ,

where R(h) = e−iλh T (h), h = t−s. Obviously, (R(h))h≥0 is a C0 -semigroup with generator Aλ = A−iλI. Then from the general theory of C0 -semigroup the above expression yields that b ∈ D(Aλ ) = D(A),

(A − iλI)b = a.

It may be noted that b is unique. By the arbitrary nature of a and λ ∈ Λ it follows that λ ∈ ρ(A). This proves the theorem. We are now in a position to formulate a main result of this section: Theorem 3.5. Λ be a closed nonemty subset of the real line. Moreover let iΛ ∩ σ(A) = .

(3.35)

Then for every f ∈ Λ(X) Eq.(3.32) has a unique (classical) bounded solution in Λ(X) provided one of the following conditions holds (i) either Λ is compact, or (ii) the operator A is bounded on X.

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In particular, the subspace Λ(X) is admissible for Eq.(3.32) in both cases. Proof. i) First of all by assumption, it is seen that the operator DΛ(X) is bounded (see e.g. p. 88 in [Levitan and Zhikov (58)]). Since Λ(X) satisfies also condition H1, by Lemma 3.7, σ(AΛ(X) ) ⊂ σ(A). (3.36) Thus, applying Theorem 1.11 to the pair of operators DΛ(X) and AΛ(X) we get the assertion of the theorem. we get the required. ii) The second case can be proved in the same manner. We now consider a more general case where the operators A and D satisfy condition P Theorem 3.6. Let (A+α) be of class Σ(θ+π/2, R) for some real α and M be a translation - invariant subspace of BU C(R, X). Moreover, let σ(A) ∩ σ(DM ) = . Then the following assertions hold true: (i) If M satisfies condition H1, then M is weakly admissible for (3.32). (ii) If M satisfies condition H2 and A is the generator of a C0 -semigroup, then M is admissible, weakly admissible and mildly admissible for (3.32). (iii) If M ⊂ AAP (X) satisfies condition H1 and A is the generator of a C0 -semigroup, then M is admissible, weakly admissible and mildly admissible for (3.32). Proof. Note that under the theorem’s assumption the operators A + α and D satisfy condition P for some real α. In fact, we can check only that sup kλR(λ, DM )k < ∞, λ∈Σ(π/2−ε,R)

where 0 < ε < π/2. Since λ ∈ Σ(π/2 − ε, R) with 0 < ε < π/2 Z ∞ kλR(λ, DM )f k = |λ|k e−λt f (· + t)dtk 0 Z ∞ ≤ |λ| e−Reλt dtkf k 0

|λ| kf k ≤ Reλ ≤ M kf k, where M is a constant independent of f . Thus, by Theorem 1.11, A

σ(DM − AM )A − α = σ(DM − AM − α) = σ(DM − (AM + α))A ⊂ σ(DM ) − σ(AM + α)

⊂ σ(DM ) − σ(AM ) − α.

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Hence A

σ(DM − AM ) ⊂ σ(DM ) − σ(A).

(3.37)

By assumption and by Lemma 3.7 since σ(DM ) ∩ σ(A) = we have σ(DM ) ∩ σ(A) = . From (3.37) and this argument we get A

0 6∈ σ(DM − AM ). Hence, this implies in particular the weak admissibility of the function space M for Eq.(3.32) proving i). Now in addition suppose that A generates a strongly continuous semigroup. Then ii) and iii) are immediate consequences of Corollary 3.6 and i). 3.2.4

Higher Order Differential Equations

In this subsection we consider the admissibility of the function space M ∩ Λ(X) where M is assumed to satisfy condition H1 and Λ is a closed subset of the real line for the equation dn u = Au + f (t), (3.38) dtn where n is a natural number. To this end, we first study the operator dn u/dtn := Dn on M ∩ Λ(X). Proposition 3.3. With the above notation the following assertions hold true: (i) n σ(DM∩Λ(X) ) ⊂ (iΛ)n .

(ii) n σ(DΛ(X) ) = (iΛ)n .

Proof.

We associate with the equation

dn u = µu + f (t), f ∈ M ∩ Λ(X) dtn the following first order equation   x01 = x2     x0 = x , 3 2 f ∈ M ∩ Λ(X). ...     0 xn = µx1 + f (t)

(3.39)

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Obviously, the unique solvability of these equations in the function space M ∩ Λ(X) are equivalent. On the other hand, by Theorem 3.5 for every f ∈ M ∩ Λ(X) Eq.(3.39) has a unique (classical) solution x(·) ∈ M ∩ Λ(X), x = (x1 , · · · , xn )T if iΛ ∩ σ(I(µ)) = , where I(µ) denotes the operator matrix associated with Eq.(3.39). A simple computation shows that σ(I(µ)) consists of all solutions to the equation tn − µ = 0. Thus, n σ(DM∩Λ(X) ) ⊂ {µ ∈ C : µ = (iλ)n for someλ ∈ Λ}.

Hence i) is proved. On the other hand, let µ ∈ Λ. Then g(·) := xeiµ· ∈ n n Λ(X). Obviously, DΛ(X) g = (iµ)n g and thus, (iµ)n ∈ σ(DΛ(X) ). Hence, ii) is proved. To proceed we make a definition Definition 3.8. The definition of admissibility for the first order equations is naturally extended to higher order equations. Observe that (iΛ)n is compact if Λ is compact. Theorem 3.7. Let Λ be a compact subset of the real line and M be a translation invariant subspace of BU C(R, X) satisfying condition H1. Moreover, let A be any closed operator in X such that σ(A) ∩ (iΛ)n = . Then for every f ∈ M ∩ Λ(X) there exists a unique (classical) solution uf ∈ M ∩ Λ(X) of Eq.(3.38). In particular, M ∩ Λ(X) is admissible for Eq.(3.38). Proof. The theorem is an immediate consequence of Theorem 3.5 and the above computation of the spectrum of D n . We recall the following notion. Definition 3.9. By a mild solution of Eq.(3.38) we understand a bounded uniformly continuous function u : R → X such that Z t1 Z t Z tn−1 dt1 dt2 ... u(s)ds ∈ D(A) 0

0

0

and

u(t) = x0 + tx1 + ...t +

Z

t

dt1 0

Z

n−1

xn−1 + A

t1

dt2 ... 0

Z

tn−1 0

Z

t

dt1 0

Z

t1

dt2 ... 0

f (s)ds (t ∈ R)

Z

tn−1

u(s)ds 0

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for some fixed x0 , x1 , ..., xn−1 ∈ X. For u ∈ BU C(R, X) we say that u is a classical solution to Eq.(3.38) if u(t) ∈ D(A), ∀t ∈ R and the n-th derivative of u (denoted by u(n) ) exists as an element of BU C(R, X) such that Eq.(3.38) holds for all t ∈ R. Remark 3.4. Obviously, a classical solution is also a mild solution. In case n = 1 if A generates a strongly continuous semigroup the above definition of mild solution on R coincides with the usual notion which we have used so far. In fact we have: 3.10. Let A be the generator of a C0 -semigroup and u satisfy RLemma t u(s)ds ∈ D(A), ∀t such that 0 Z t Z t f (s)ds. (3.40) u(s)ds + u(t) = u(0) + A 0

0

Then u satisfies

u(t) = T (t − s)u(s) +

Z

t s

T (t − ξ)f (ξ)dξ, ∀t ≥ s

where T (t) = etA . Conversely, if u satisfies Eq.(3.41), then D(A), ∀t and u satisfies Eq.(3.40).

(3.41) Rt 0

u(s)ds ∈

Proof. Suppose that u is a solution to Eq.(3.40). Then, we will show that it is also a solution to Eq.(3.41). In fact, without loss of generality we verify that u satisfies Eq.(3.41) for s = 0. To this purpose let us define the function Z t T (t − ξ)f (ξ)dξ, t ≥ 0. w(t) = T (t)u(0) + 0

We now show that w satisfies Eq.(3.40) for t ≥ 0 as well. In fact, using the following facts from semigroup theory Z t T (t)x − x = A T (s)xds, ∀x ∈ X, 0

and the following which can be verified directly by definition Z t Z t A T (s − η)f (η)ds = lim(1/h)(T (h) − I) T (s − η)f (η)ds h↓0

η

η

= T (t − η)f (η) − f (η) we have A

Z

t 0

w(s)ds = T (t)u(0) − u(0) + A

Z tZ 0

s 0

T (s − η)f (η)dη.

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By a change of order of integrating we get Z tZ s Z t Z t A T (s − η)f (η)dη = A dη T (s − η)f (η)ds. 0

0

0

η

By the above facts we have Z t Z t Z t Z t dηA T (s − η)f (η)ds = T (t − η)f (η)dη − f (η)dη. 0

η

0

0

This shows that w satisfies Eq.(3.40) for all t ≥ 0. Define g(t) = w(t)−u(t). Obviously, Z t g(s)ds, ∀t ≥ 0. g(t) = A 0

Since A generates a strongly continuous semigroup the Cauchy problem x0 = Ax, x(0) = 0 ∈ D(A)

has a unique solution zero. Hence, u(t) = w(t), ∀t ≥ 0, i.e. u(t) satisfies Eq.(3.41) for all t ≥ 0.

By reversing the above argument we can easily show the converse. Hence, the lemma is proved. Lemma 3.11. Let A be a closed operator and u be a mild solution of Eq.(3.38) and φ ∈ L1 (R) such that its Fourier transform has compact support. Then φ ∗ u is a classical solution to Eq.(3.38) with forcing term φ ∗ f . Proof.

Let us define Z t Z t f (s)ds, t ∈ R, u(s)ds, F1 (t) = U1 (t) = 0 0 Z t Z t Uk (t) = Uk−1 (s)ds, Fk (t) = Fk−1 (s)ds, t ∈ R, k ∈ N. 0

0

Then, by definition, we have

u(t) = Pn (t) + A(Un (t)) + Fn (t), t ∈ R,

where Pn is a polynomial of order of n − 1. From the closedness of A, we have u ∗ φ(t) = Pn ∗ φ(t) + A(Un ∗ φ(t)) + Fn ∗ φ(t), t ∈ R. Since the Fourier transform φ has compact support all convolutions above are infinitely differentiable. From the closedness of A we have that (Un ∗ φ)(k) (t) ∈ D(A), t ∈ R and A((Un ∗ φ)(k) (t)) =

dk A(Un ∗ φ(t)), t ∈ R. dtk

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(k)

Set Vn (t) = Un ∗φ(t). Since Un (t) = Un−k (t), k = 0, 1, 2 · · · , n and U0 (t) = (n) u(t) we have V (k) (t) = Un−k ∗ φ(t), k = 0, 1, 2, · · · , n and Vn (t) = u ∗ φ(t). Hence, Un ∗ φ(t) = Vn (t) Z t n−1 X tk 1 Un−k ∗ φ(0) + = (t − s)n−1 u ∗ φ(s)ds. k! (n − 1)! 0 k=0

Since Un ∗ φ(t) ∈ D(A), Un−k ∗ φ(0) ∈ D(A), k = 1, 2, · · · , n − 1 it follows that the integral above belongs also to D(A). Furthermore, we can check that Z t 1 n−1 (t − s) u ∗ φ(s)ds u ∗ φ(t) = Pn ∗ φ(t) + Qn (t) + A (n − 1)! 0 Z t 1 + (t − s)n−1 f ∗ φ(s)ds, (n − 1)! 0

where Pn , Qn are polynomials of order of n − 1 which appears when one expands A(Un ∗ φ(t)) and Fn ∗ φ(t), respectively. Now, since all functions in the above expression are infinitely differentiable, Pn , Qn are polynomials of order of n − 1 and A is closed we can differentiate the expression to get dn (u ∗ φ)(t) = A(u ∗ φ(t)) + f ∗ φ(t), ∀t ∈ R. dtn This proves the lemma. We now recall the notion of Λ-class of functions. Definition 3.10. A translation invariant subspace F ⊂ BU C(R, X) is said to be a Λ-class if and only if it satisfies (1) (2) (3) (4)

F F F F

is a closed subspace of BU C(R, X) ; contains all constant functions; satisfies condition H1; is invariant by multiplication by eiξ· , ∀ξ ∈ R.

Let F be a Λ-class and u be in BU C(R, X). Then, by definition

spF (u) := {ξ ∈ R : ∀ε > 0∃f ∈ L1 (R)

such that suppFf ⊂ (ξ − ε, ξ + ε) and f ∗ u 6∈ F}.

Lemma 3.12. If f ∈ F , where F is a Λ-class, then ψ∗f ∈ F, ∀ψ ∈ L1 (R) such that the Fourier transform of ψ has compact support. Proof.

For the proof we refer the reader to p.60 in [Basit (12)].

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Hence, Theorem 3.7 yields the following: Theorem 3.8. Let F be a Λ-class, A be a closed linear operator with nonempty resolvent set. Then for any mild solution u to Eq.(3.38) with f ∈ F, spF (u) ⊂ {λ ∈ R : (iλ)n ∈ σ(A)}.

(3.42)

Proof. Let λ0 ∈ R such that (iλ0 )n 6∈ σ(A). Then, since σ(A) is closed there is a positive number δ such that for all λ ∈ (λ0 − 2δ, λ0 + 2δ) we have (iλ)n 6∈ σ(A). Let us define Λ := [λ0 − δ, λ0 + δ]. Then by Theorem 3.7 for every y ∈ Λ(X) ∩ F there is a unique (classical) solution x ∈ Λ(X) ∩ F. Let ψ ∈ L1 (R) such that suppFψ ⊂ Λ. Put v := ψ ∗ u, g := ψ ∗ f . Then, by Lemma 3.12 g ∈ F and by Proposition 2.5 in [Basit (12)] spF (g) ⊂ suppFψ ∩ spF (f ) ⊂ Λ. Thus g ∈ Λ(X) ∩ F. Since spF (v) ⊂ Λ by Theorem 3.7 we see that Eq.(3.38) has a unique solution in Λ(X) which should be v. Moreover, applying again Theorem 3.7 we can see that the function v should belong to Λ(X) ∩ F. We have in fact proved that λ0 6∈ spF (u). Hence the assertion of the theorem has been proved. In a standard manner we get the following: Corollary 3.7. Let F be a Λ-class, σ(A) ∩ (iR)n be countable. Moreover, let u be such a mild solution to Eq.(3.38) that Z 1 t −iλs e u(x + s)ds lim t→∞ t 0 exists for every λ ∈ spF (u) uniformly with respect to x ∈ R. Then u ∈ F. Proof. The corollary is an immediate consequence of Theorem 1.18 and Theorem 3.8. In particular, we can take F = AP (X), AAP (X) and get spectral criteria for almost periodicity and asymptotic almost periodicity for solutions to the higher order equations (3.38). Next, we consider the admissibility of a given translation invariant closed subspace M for the higher order equation (3.38). Since the geometric properties of the set (iR)n play an important role, we consider here only the case n = 2, i.e., the following equation d2 u = Au + f (t). (3.43) dt2 It turns out that for higher order equations conditions on A are much weaker than for the first order ones. Indeed, we have

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Theorem 3.9. Let A be a linear operator on X such that there are positive constants R, θ and Σ(θ, R) ⊂ ρ(A) and

sup λ∈Σ(θ,R)

|λ|kR(λ, A)k < ∞.

Furthermore, let M be a translation invariant closed subspace of BU C(R, X) which satisfies condition H1 such that 2 σ(DM ) ∩ σ(A) = .

Then M is admissible for the second order equation (3.43). 2 Proof. We will apply Theorem 1.11 to the pair of linear operators DM and AM . To this end, by Proposition 3.3 we observe that 2 σ(DM ) ⊂ (iR)2 = (−∞, 0].

On the other hand, for 0 < ε < θ we can show that there is a constant M such that the following estimate holds 2 kR(λ, DM )k ≤

M , ∀λ 6= 0, |arg(λ) − π| < ε. |λ|

In fact, this follows immediately from well known facts in Chapter 2 in [Daleckii and Krein (24)]. To make it more clear, we consider the first order 2 ) the equation of the form (3.39) for the case n = 2. For every λ ∈ ρ(DM associated equation has an exponential dichotomy and its Green function 2 is nothing but R(λ, DM ). Furthermore, since M is translation invariant 2 note that D(DM ) is dense in M. Thus, applying Theorem 1.11 to the pair 2 of operators DM , AM we have 2 − A ). 0 ∈ ρ(DM M 2 It remains to show that for every f ∈ M0 := D(DM ) there is a unique classical solution u on R. In fact, denoting 2 − A )−1 , G := (DM M 2 2 we can easily see that since DM , AM commute with DM , so does G. By 2 definition, for λ ∈ ρ(DM ) , since G is bounded on M 2 2 GR(λ, DM ) = R(λ, DM )G. 2 Hence there is g ∈ M such that f = R(λ, DM )g. Thus, by the above 2 2 equality Gf = R(λ, DM )Gg ∈ D(DM ). This shows the admissibility of M for Eq.(3.43).

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3.2.5

Abstract Functional Differential Equations

This subsection will be devoted to some generalization of the method discussed in the previous ones for functional differential equations of the form dx(t) = Ax(t) + [Bx](t) + f (t), ∀t ∈ R, (3.44) dt where the operator A is a linear operator on X and B is assumed to be an autonomous functional operator. We first precise the notion of autonomousness for functional operators B: Definition 3.11. Let B be an operator, everywhere defined and bounded on BU C(R, X) into itself. B is said to be an autonomous functional operator if for every φ ∈ BU C(R, X) S(τ )Bφ = BS(τ )φ, ∀τ ∈ R, where (S(τ ))τ ∈R is the translation group S(τ )x(·) := x(τ +·) in BU C(R, X). In connection with autonomous functional operators we will consider closed translation invariant subspaces M ⊂ BU C(R, X) which satisfy condition H3. Recall that if B is an autonomous functional operator and M satisfies condition H3, then by definition, M is left invariant under B. Definition 3.12. Let A be the generator of a C0 -semigroup and B be an autonomous functional operator. A function u on R is said to be a mild solution of Eq.(3.44) on R if Z t (t−s)A u(t) = e u(s) + e(t−ξ)A [(Bu)(ξ) + f (ξ)]dξ, ∀t ≥ s. s

As we have defined the notion of mild solutions it is natural to extend the notion of mild admissibility for Eq.(3.44) in the case where the operator A generates a strongly continuous semigroup. It is interesting to note that in this case because of the arbitrary nature of an autonomous functional operator B nothing can be said on the “well posedness” of Eq.(3.44). We refer the reader to Chapter 1 for particular cases of “finite delay” and “infinite delay” in which Eq.(3.44) is well posed.) However, as shown below we can extend our approach to this case. Now we formulate the main result for this subsection.

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Theorem 3.10. Let A be the infinitesimal generator of an analytic strongly continuous semigroup, B be an autonomous functional operator on the function space BU C(R, X) and M be a closed translation invariant subspace of AAP (X) which satisfies condition H3. Moreover, assume that σ(DM ) ∩ σ(A + B) = . Then M is mildly admissible for Eq. (3.44), i.e., for every f ∈ M there is a unique mild solution uf ∈ M of Eq.(3.44). Proof. Since M satisfies condition H3, for every f ∈ M we have Bf ∈ M. Thus, D((A + B)M ) = {f ∈ M : Af (·) + Bf ∈ M} = {f ∈ M : Af (·) ∈ M} = D(AM ).

Hence (A + B)M = AM + BM . As M satisfies condition H3 it satisfies condition H1 as well. Thus, by Lemma 3.7, σ(AM ) ⊂ σ(A) ⊂ σ(A) and kR(λ, AM )k ≤ kR(λ, A)k, ∀λ ∈ ρ(A). Since B is bounded DM and (A + B)M = AM + BM satisfy condition P. From Lemma 2 and the remarks follows in [Naito and Minh (74)] it may be seen that AM is the infinitesimal generator of the strongly continuous semigroup (T (t))t≥0 T (t)f (ξ) := etA f (ξ), ∀f ∈ M, ξ ∈ R. Hence D((A + B)M ) = D(AM ) is dense everywhere in M. It may be noted that R(λ, A + B) commutes with the translation group. Since M satisfies condition H3 we can easily show that σ((A + B)M ) ⊂ σ(A + B). Applying Theorem 1.11 we get σ(DM − (A + B)M ) ⊂ σ(DM ) − σ((A + B)M ). Hence 0 ∈ ρ(DM − (A + B)M ).

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On the other hand, since BM is bounded on M DM − (A + B)M = DM − AM − BM = LM − BM

we have 0 ∈ ρ(LM − BM ).

(3.45)

If u, f ∈ M such that (LM − BM )u = f , then LM u = BM u + f. By definition of the operator LM , this is equivalent to the following Z t (t−s)A u(t) = e u(s) + e(t−ξ)A [(BM u)(ξ) + f (ξ)]dξ, ∀t ≥ s, s

i.e., u is a mild solution to Eq.(3.44). Thus (3.45) shows that M is mildly admissible for Eq.(3.44). Remark 3.5. Sometime it is convenient to re-state Theorem 3.10 in other form than that made above. In fact, in practice we may encounter difficulty in computing the spectrum σ(A+B). Hence, alternatively, we may consider D −A−B as a sum of two commuting operators D −B and A if B commutes with A. In subsection 3.4 we again consider this situation. 3.2.6

Examples and Applications

In this subsection we will present several examples and applications and discuss the relation between our results and the previous ones. As typical examples of the function spaces Λ(X) , where Λ is a closed subset of the real line we will take the following ones: Example 1 The space of all X valued continuous τ -periodic functions P(τ ). In this case Λ = {2kπ/τ, k ∈ Z}. Example 2 Let Λ be a discrete subset of R. Then Λ(X) will consists of almost periodic functions. Example 3 Let Λ be a countable subset of R. Then Λ(X) will consists of almost periodic functions if in addition one assumes that X does not contain any subspace which is isomorphic to the space c0 (see [Levitan and Zhikov (58)]).

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Below we will revisit one of the main results of [Langenhop (53)] to show how our method fits in the problem considered in [Langenhop (53)]. Moreover, our method can be easily extended to the infinite dimensional case. Example 4 (cf. [Langenhop (53)]) Consider the following ordinary functional differential equation Z ∞ x0 (t) = [dE(s)]x(t − s) + f (t), x ∈ Cn , t ∈ R, (3.46) 0

where E is an n × n matrix function with elements in C, f is a Cn -valued almost periodic function. In addition, we assume that E is continuous from the left and of bounded total variation on [0, ∞) , i.e. Z ∞ 0<γ= |dE(s)| < ∞. 0

As is well known for every f ∈ AP (Cn ) there is a corresponding Fourier series ∞ X ak eiλk t . k=0

We define

A0q := {f ∈ AP (Cn ) : a0 = 0, |λk | ≥ q, k = 1, 2, · · · }

and Aq = A0q + Vc , where Vc is the set of all Cn -valued constant functions. Now we define our operator Z ∞ Bu(t) := [dE(s)]u(t − s), t ∈ R, u ∈ AP (Cn ). 0

Obviously, B is an autonomous functional operator with kBk ≤ γ. If we define Λ := {η ∈ R : |η| ≥ q} , then A0q = AP (Cn ) ∩ Λ(Cn ). Now we prove the following: Assertion 1 Under the above notations and assumptions Eq.(3.46) has a unique almost periodic solution xf ∈ A0q for every f ∈ A0q if γ < q. Proof. In fact, by assumption it is obvious that the spectral radius rσ (B) < γ. Hence, iΛ ∩ σ(B) = . If in addition we assume that M :=

Z

∞ 0

dE(s)

(3.47)

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is a nonsigular matrix, then Assertion 1 implies the following: Assertion 2 Under Assertion 1’s assumptions and the nonsingularity of the matrix (3.47) there exists a unique solution xf ∈ Aq to Eq.(3.46) for every f ∈ Aq . Proof. In this case the operator d/dt − B is a direct sum of two invertible operators in A0q and Vc . Remark 3.6. In Section 3 in [Langenhop (53)] Assertion 2 has been proved with a little stronger assumption, namely, γδ < q , where δ ≥ 1 is an “absolute constant” (in terminology of p.401 in [Langenhop (53)]). The condition γ < q of Assertion 2 becomes also necessary in many cases. To show this, we consider the case Bu(t) = Bu(t + τ ), ∀t ∈ R, u ∈ BU C(R, X), where τ is a given constant, B is a matrix. Now suppose that there exists a unique solution xf ∈ Aq to Eq.(3.46) for every f ∈ Aq . Denoting Gf := xf we see that G is a bounded linear operator on Aq . Moreover, since B commutes with translation group so does G , i.e., DGf = GDf, ∀f ∈ D(D). Taking f := eiλt y we have Dxf = DGf = GDf = λGf = λxf . Hence, xf (t) = eiλt x for some x. Substituting this into Eq.(3.46) we get the assertion that given |λ| ≥ q for every y ∈ Cn there exists a unique x ∈ Cn such that iλx − eiτ λ Bx = y. This shows that iλ ∈ ρ(eiτ λ B) = eiτ λ ρ(B) and yields γ < q. In the following example we will revisit a problem discussed in [Sljusarcuk (98)] with an unbounded A. Example 5 Let us consider the equation N

X dx(t) = Ax(t) + Bk x(t + τk ) + f (t), t ∈ R, dt

(3.48)

k=1

where A is the infinitesimal generator of an analytic C0 -semigroup, Bk , k = 1, · · · , N are bounded linear operators on X which are commutative with each other and A , τk , k = 1, · · · , N are given reals and f is a bounded uniformly continuous function. We denote Λ = sp(f ).

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Assertion 3 Let Λ be bounded. Then if σ(A) ∩ ∪λ∈Λ σ(iλ −

N X k=1

Bk )eiτk λ ) = ,

Eq.(3.48) has a unique classical solution in Λ(X). Proof.

If Λ is bounded , then DΛ(X) is bounded. Hence, if σ(DΛ(X) −

N X k=1

Bk eτk DΛ(X) ) ∩ σ(A) =

Eq.(3.48) has a unique classical solution in Λ(X). In turn, using the estimates of spectra as in [Sljusarcuk (98)] we get σ(DΛ(X) −

N X k=1

Bk eτk DΛ(X) ) ⊂ ∪λ∈Λ σ(iλ −

N X

Bk eiτk λ ).

k=1

Assertion 4 Let f be almost periodic and Bk , k = 1, · · · , N be commutative with each other and A and iΛ ∩ (σ(A) + ∪λ

k ∈e

iτk Λ

σ(

N X k=1

Bk λk )) = .

Then Eq.(3.48) has a unique almost periodic mild solution in Λ(X). Proof. First using the Weak Spectral Mapping Theorem (see e.g. [Nagel (73)]) we have σ(S(τk )) = eiτk Λ . In view of Theorem 1 in [Sljusarcuk (98)], denoting the multiplication operator by Bk by also Bk for the sake of simplicity, we have σ(

N X k=1

Bk S(τk )) ⊂ ∪λ

k ∈e

iτk Λ

σ(

N X

Bk λk ).

k=1

By the commutativeness assumption applying Theorem 3.10 and then Theorem 1.11 we get the conclusion of the assertion. In case A is the generator of a C0 -semigroup which is not necessarily analytic we can still apply Theorem 3.2 and the commutativeness of the operators A, B as shown in the following example:

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Example 6 Let A be the infinitesimal generator of a strongly continuous semigroup of linear operators on X, B be an autonomous functional operator on BU C(R, X) and M be a translation invariant subspace of AAP (X). Moreover, we assume that B and A commute. The only difference between this example and the previous one is that the semigroup generated by A may not be analytic. However, we can find conditions for the admissibility of M by using evolution semigroup associated with A as in Theorem 3.2. In fact, in M, since BM is bounded it generates the norm continuous semigroup (B h )h≥0 . Hence, −DM + AM + BM generates a strongly continuous semigroup (T h B h )h≥0 . Thus, in view of spectral inclusion of strongly continuous semigroups, this generator is invertible if 1 6∈ σ(T 1 B 1 ). Using the commutativeness of the operators under consideration and the Weak Spectral Mapping Theorem for the translation group on M we have σ(T 1 B 1 ) ⊂ σ(T 1 ).σ(B 1 ) ⊂ e−DM σ(eA ).σ(B 1 ).

(3.49)

Hence the following is obvious: Assertion 5 If then

1 6∈ e−DM σ(eA ).σ(B 1 ),

dx(t) = Ax(t) + [Bx](t) + f (t), dt has a unique mild solution in M for every given f ∈ M.

(3.50)

As an application suppose that we are given an almost periodic function f . Let M ⊂ AP (X) consisting of all functions g such that sp(g) ⊂ sp(f ). Then the above condition can be written as 1 6∈ e−isp(f ) σ(eA ).eσ(BM )

(3.51)

1 6∈ σ(eA )eb .

(3.53)

which implies the existence of an almost periodic mild solution to Eq.(3.49). To illustrate the usefulness of (3.51) we consider the following case of Eq.(3.50) dx(t) = Ax(t) + bx(t + 1) + f (t), (3.52) dt where b ∈ R and f is 1-periodic and continuous. In this case, B = bS(1). Hence, sp(f ) = 2πZ and condition (3.51) can be written as Hence, if condition (3.53) holds true, then Eq.(3.52) has a unique 1-periodic mild solution.

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103

Decomposition Theorem and Periodic, Almost Periodic Solutions

We begin this section with several classical examples showing that in the resonant case when the conditions of the type σ(A) ∩ iR 6= ∅ are not satisfied, the problem of finding almost periodic solutions becomes much more complicated. Example 3.4. Consider the simplest form of inhomogeneous equations when A = 0. In this case, actually we are concerned with conditions for the integral Z t F (t) := f (ξ)dξ, t ∈ R 0

is τ -periodic, where f is assumed to be a scalar τ -periodic continuous function. A simple computation shows that F is of the form F (t) = mt + G(t), where Z 1 τ f (ξ)dξ τ 0 G is a τ -periodic function.

m :=

Thus F is τ -periodic if and only if m = 0, or equivalently, F is bounded on R (actually on any half line). That is, a boundedness condition is needed. In the infinite dimensional case, the boundedness condition of the above type is not sufficient. Further conditions on the geometry of the Banach space, in which the equation is studied, is needed. Example 3.5. Let c0 be the Banach space of numerical sequences ξ := {ξn }n∈N ⊂ C such that limn→∞ ξn = 0 with kξk := supn∈N |ξn |. Let Z t f (t) := {1/n cos t/n}n∈N , F (t) = f (η)dη = {sin t/n}n∈N . 0

One can shows that the range of the function F is not precompact, and so the function F cannot be almost periodic. In fact, suppose that the range of F is precompact. Then, consider the sequence of functionals φn ∈ c∗0 defined by the formula φn (ξ) = ξn if ξ = {ξn }n∈N . If the range R(F ) of F is precompact, then the convergence φn (ξ) would be uniform for ξ ∈ R(F ). But φn (F (t)) = sin(t/n) → 0 non-uniformly with respect to t ∈ R. This contradicts the assumption on the precompactness of R(F ).

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To see how the resonance influences on the amplitude of oscillation we consider the following example. Example 3.6. Consider the equation t ∈ R, x(t) ∈ R.

x ¨(t) + x(t) = sin t,

One can show that the function x1 (t) = − 2t cos t is a solution of this equation. Below is the graph of this solution on the interval [−60, 60] which exhibits its unboundedness of the amplitude as time tends to infinity.

30

20 y 10

-60

-40

-20

0

20

40

60

x

-10

-20

-30

And any solution of the above equation is of the form x(t) = C1 cos t + C2 sin t + x1 (t). Hence any solution of the equation is unbounded, so it is not almost periodic. We consider in this section the following linear inhomogeneous integral equation Z t x(t) = U (t, s)x(s) + U (t, ξ)g(ξ)dξ, ∀t ≥ s; t, s ∈ R, (3.54) s

where f is continuous, x(t) ∈ X, X is a Banach space, (U (t, s))t≥s is assumed to be a 1-periodic evolutionary process on X. As is known, continuous solutions of Eq.(3.54) correspond to the mild solutions of evolution equations dx = A(t)x + f (t), t ∈ R, x ∈ X, dt

(3.55)

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where A(t) is a (in general, unbounded) linear operator for every fixed t and is 1-periodic in t, and (U (t, s))t≥s is generated by Eq.(3.55). In the previous sections we have studied conditions for the existence and uniqueness (in some classes of function spaces) of almost periodic solutions of Eq.(3.54). In fact we have shown that if the following nonresonant condition holds (σ(P ) ∩ S 1 ) ∩ eisp(f ) = ,

(3.56)

where P := U (1, 0), S 1 denotes the unit circle of the complex plane, and f is almost periodic, then there exists an almost periodic solution xf to Eq.(3.54) which is unique if one requires eisp(xf ) ⊂ eisp(f ) . We may ask a question as what happens in the resonant case where condition (3.56) fails. Historically, this question goes back to a classical result of ordinary differential equations saying that supposing the finite dimension of the phase space X and the 1-periodicity of f Eq.(3.54) has a 1-periodic solution if and only if it has a bounded solution (see e.g. Theorem 20.3, p. 278 in [Amann (1)]). It is the purpose of this section to give an answer to the general problem as mentioned above (Massera-type problem): Let Eq.(3.54) have a bounded (uniformly continuous) solution xf with given almost periodic forcing term f . Then, when does Eq.(3.54) have an almost periodic solution w (which may be different from xf ) such that eisp(w) ⊂ eisp(f )

?

In connection with this problem we note that various conditions are found on the bounded solution, itself, and the countability of the part of spectrum σ(P )∩S 1 so that the bounded solution itself is almost periodic, or more generally, together with f belongs to a given function space F. Here we note that this philosophy in general does not apply to the Masseratype problem for almost periodic solutions. In fact, it is not difficult to give a simple example in which f is 1-periodic and a bounded (uniformly continuous) solution to Eq.(3.54) exists, but this bounded solution itself is not 1-periodic. Our method is to use the evolution semigroup associated with the process (U (t, s))t≥s to study the harmonic analysis of bounded solutions to Eq.(3.54). As a result we will prove a spectral decomposition theorem for bounded solutions (Theorem 3.12 and Theorem 3.13) which seems to be useful in dealing with the above Massera-type problem. In fact, we will

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apply the spectral decomposition theorem to find new spectral criteria for the existence of almost periodic solutions and will consider particular cases to show the usefulness of this spectral decomposition technique. More concretely, even in the case where condition (3.56) fails we can still prove the existence of a bounded uniformly continuous solution w to Eq.(3.54) such that eisp(w) = eisp(f ) provided that (σ(P ) ∩ S 1 )\eisp(f ) is closed, and that Eq.(3.54) has a bounded uniformly continuous solution u (Corollary 3.8). Since w is a ”spectral component” of u in case u is almost periodic the Fourier series of w is part of that of u (Corollary 3.9). Our Corollary 3.10 will deal with a particular autonomous case in which Corollary 3.8 fails to give a spectral criterion for the existence of quasi-periodic mild solutions. For the sake of simplicity of notations we will use throughout the section the following notation: σ(g) := eisp(g) for every bounded uniformly continuous function g. Throughout the section we will denote by σΓ (P ) = σ(P ) ∩ S 1 . Throughout this section (U (t, s))t≥s will be assumed to be a 1-periodic strongly continuous evolutionary process. The operator U (1, 0) will be called the monodromy operator of the evolutionary process (U (t, s))t≥s and will be denoted by P throughout this section. Note that the period of the evolutionary processes is assumed to be 1 merely for the sake of simplicity. Recall that Definition 3.13. Let (U (t, s))t≥s , (t, s ∈ R) be a 1-periodic strongly continuous evolutionary process and F be a closed subspace of BU C(R, X) such that for every fixed h > 0, g ∈ F the map t 7→ U (t, t − h)g(t − h) belongs to F. Then the semigroup of operators (T h )h≥0 on F, defined by the formula T h g(t) = U (t, t − h)g(t − h), ∀t ∈ R, h ≥ 0, g ∈ F,

is called evolution semigroup associated with the process (U (t, s))t≥s on F. We refer the reader to the previous section for further information on this semigroup. In the case where the evolution semigroup (T h )h≥0 is strongly continuous on F the explicit formula for the generator A of (T h )h≥0 is as follows: Lemma 3.13. Let (T h )h≥0 be strongly continuous on F, a closed subspace of BU C(R, X). Then its generator A is the operator on F with D(A) consisting of all g ∈ F such that g is a solution to Eq.(3.54) with some f ∈ F (in this case such a function f is unique), by definition, Ag = −f . Proof.

See the proof of Lemma 3.1.

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Spectral Decomposition

Let us consider the subspace M ⊂ BU C(R, X) consisting of all functions v ∈ BU C(R, X) such that eisp(v) := σ(v) ⊂ S1 ∪ S2 ,

(3.57)

where S1 , S2 ⊂ S1 are disjoint closed subsets of the unit circle. We denote by Mv = spann{S(t)v, t ∈ R}, where (S(t))t∈R is the translation group on BU C(R, X) , i.e. S(t)v(s) = v(t + s), ∀t, s ∈ R. Theorem 3.11. Under the above notation and assumptions the function space M can be split into a direct sum M = M1 ⊕ M2 such that v ∈ Mi if and only if σ(v) ⊂ Si for i = 1, 2. Proof.

Let v ∈ M . Then, as is known (see Chapter 1 ) isp(v) = σ(DMv ).

(3.58)

Thus, by the Weak Spectral Mapping Theorem (see Chapter 1) σ(S(1)|Mv ) = eσ(DMv ) = σ(v) ⊂ S1 ∪ S2 .

(3.59)

Hence there is a spectral projection in Mv (note that in general we do not claim that this projection is defined on the whole space M) Z 1 R(λ, S(1)|Mv )dλ , Pv1 := 2iπ γ where γ is a contour enclosing S1 and disjoint from S2 , (or in general a union of fintely many such countours) by which we have σ(Pv1 S(1)Pv1 ) ⊂ S1 .

(3.60)

On the other hand, denote Λi ⊂ BU C(R, X) consisting of all functions u such that σ(u) ⊂ Si for i = 1, 2 . Then obviously, Λi ⊂ M . Moreover, they are closed subspaces of M, Λ1 ∩ Λ2 = {0} . Now we show that if v ∈ M, then Pv1 v ∈ Λ1 and v − v1 := v2 ∈ Λ2 . If this is true, then it yields that M = Λ 1 ⊕ Λ2 . To this end, we will prove σ(vj ) ⊂ Sj , ∀j = 1, 2.

(3.61)

In fact we show that Mv1 = ImPv1 . Obviously, in view of the invariance of ImPv1 under translations we have Mv1 ⊂ ImPv1 . We now show the inverse. To this end, let y ∈ ImPv1 ⊂ Mv . Then, by definition, we have y = lim xn , n→∞

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where xn can be represented in the form N (n)

xn =

X

k=1

αk,n S(tk,n )v, αk,n ∈ C, tk,n ∈ R ∀n.

Hence, since y, xn ∈ Mv N (n)

y = Pv1 y = lim

n→∞

X

αk,n S(tk,n )Pv1 v

X

αk,n S(tk,n )v1 .

k=1

N (n)

= lim

n→∞

(3.62)

k=1

This shows that y ∈ Mv1 . Thus, by the Weak Spectral Mapping Theorem and (3.60), eisp(v1 ) = σ(S(1)|Mv1 ) = σ(S(1)|ImPv1 ) ⊂ S1 . By definition, v1 ∈ Λ1 and similarly, v2 ∈ Λ2 . Thus the theorem is proved.

Remark 3.7. Below for every v ∈ M we will call vj , j = 1, 2, as defined in the proof of Theorem 3.11, spectral components of the functions v. It is easily seen that if in the proof of Theorem 3.11, v is assumed to be almost periodic, then both spectral components vj are almost periodic. We will need the following lemma in the sequel Lemma 3.14. Let f be in BU C(R, X) and (U (t, s))t≥s be a 1-periodic strongly continuous evolutionary process. Then the following assertions hold: (i) If T : R → L(X) be 1-periodic and strongly continuous, then σ(T (·)f (·)) ⊂ σ(f ). (ii) If f is of precompact range, then Z t+1 U (t + 1, ξ)f (ξ)dξ ⊂ σ(f ). σ t

Proof. (i) Let Tn be the nth Cesaro mean of the Fourier series of T , so Tn is 1-periodic trignometric polynomial with value in L(X) and kTn (s)k ≤ sup0≤t≤1 kT (t)k and Tn (s)x → T (s)x uniformly in s for fixed x ∈ X. For every n it is easily seen that σ(Tn (·)f (·)) ⊂ σ(f ). Set Λ := {λ ∈ R : eiλ ∈

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σ(f )}. Obviously, Λ is closed and sp(Tn (·)f (·) ⊂ Λ. Thus, if φ ∈ L1 (R) ˆ ∩ Λ = , then and (suppφ) Z ∞ Z ∞ 0= φ(t − s)Tn (s)f (s)ds → φ(t − s)T( s)f (s)ds −∞

−∞

as n → ∞, by Dominated Convergence Theorem. Thus Z ∞ φ(t − s)T( s)f (s)ds = 0 −∞

for all such φ. This proves (i). (ii) Since f is of precompact range the evolution semigroup (T h )h≥0 associated with the process (U (t, s))t≥s is strongly continuous at f . Thus, in view of (i) ! Z h ξ σ T f dξ ⊂ (f ), ∀h ≥ 0. 0

On the other hand Z Z 1 T ξ f dξ(t+1) = 0

1

U (t+1, t+1−ξ)f (t+1−ξ)dξ = 0

Z

t+1

U (t+1, η)f (η)dη. t

This proves (ii).

Lemma 3.15. Let u be a bounded uniformly continuous solution to (3.54) and f be of precompact range. Then the following assertions hold true: (i) σ(u) ⊂ σΓ (P ) ∪ σ(f ),

(3.63)

σ(u) ⊃ σ(f ) .

(3.64)

(ii)

Proof.

(i) Set P (t) := U (t, t − 1), ∀t ∈ R, G := {λ ∈ C : eλ ∈ ρ(P )}. Z t+1 U (t + 1, ξ)f (ξ)dξ, t ∈ R. g(t) := t

By Lemma 3.14 σ(g) ⊂ σ(f ). By the definition of Carleman spectrum, R ∞ −λt    0 e u(t)dt, (Reλ > 0); u ˆ(λ) =   − R ∞ eλt u(−t)dt, (Reλ < 0). 0

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110

Hence, for Reλ > 0 and λ ∈ G we have Z ∞ u ˆ(λ) = e−λt u(t)dt, Z0 ∞ = e−λt R(eλ , P (t))(eλ − P (t))u(t)dt 0

=

Z

1

e−λt eλ R(eλ , P (t))u(t)dt

0

Z

∞

e−λt R(eλ , P (t))(u(t + 1) − P (t)u(t))dt Z ∞ e−λt R(eλ , P (t))g(t)dt, = H(λ) + +

0

(3.65)

0

where

H(λ) :=

Z

1

e−λt eλ R(eλ , P (t))u(t)dt. 0

Obviously, H(λ) is analytic in G. On the other hand, since R(eλ , P (t)) is 1periodic strongly continuous (see Lemma 3.3), by Lemma 3.14 the function g1 (t) := R(eλ , P (t))g(t) has the property that σ(g1 ) ⊂ σ(f ). Thus from (3.65), for Reλ > 0, λ ∈ G, u ˆ(λ) = H(λ) + gˆ1 (λ).

(3.66)

ζ0

Finally if ζ0 ∈ R : e 6∈ σΓ (P ) ∪ σ(f ), then u ˆ has an analytic continuation at ζ0 . This completes the proof of (i). (ii) Under the assumptions it may be seen that the evolution semigroup (T h )h≥0 associated with (U (t, s))t≥s is strongly continuous at the functionu ∈ BU C(R, X) (this can be checked directly using Eq.(3.54)), and f . Hence, by Lemma 3.13 lim+

h→0

T hu − u = Au = −f . h

(3.67)

Hence, to prove (3.64) it suffices to show that σ(T h u) ⊂ σ(u) . In turn, this is clear in view of Lemma 3.14. We are now in a position to state the main result of this section. Theorem 3.12. (Spectral Decomposition Theorem) Let u be a bounded, uniformly continuous solution to Eq.(3.54). Moreover, let f have precompact range and the sets σ(f ) and σ(P ) ∩ S 1 be contained in a disjoint union of the closed subsets S1 , · · · , Sk of the unit circle. Then the solution u can be decomposed into a sum of k spectral components

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uj , j = 1, · · · , k such that each uj , j = 1, · · · , k is a solution to Eq.(3.54) Pk with f = fj , j = 1, · · · , k, respectively, where f = j=1 fj is the decomposition of f into the sum of spectral components as described in Theorem 3.11, Pk i.e. u = j=1 uj , σ(uj ), σ(fj ) ⊂ Sj , j = 1, · · · , k and uj ∈ BU C(R, X) is a solution to Eq.(3.54) with f := fj for j = 1, · · · , k. Proof. Let us denote by N the subspace of BU C(R, X) consisting of all functions u such that σ(u) ⊂ ∪kj=1 Sj . Then, by assumptions and Theorem 3.11 there are corresponding spectral projections P1 , · · · , Pk on N with properties that (1) Pj Pn = 0 if j 6= n, (2) Σkj=1 Pj = I, (3) If u ∈ ImPj , then σ(Pj u) ⊂ Sj for all j = 1, · · · , k . Note that by Lemma 3.15 for every positive h and j = 1, · · · , k the operator T h leaves ImPj invariant. Hence, N and ImP1 , · · · , ImPk are invariant under the semigroup (T h )h≥0 . Consequently, since u is a solution to Eq.(3.54) and f has precompact range the evolution semigroup (T h )h≥0 is strongly continuous at u and f . Using the explicit formula for the generator of (T h )h≥0 as described in Lemma 3.13 we have Pj f = Pj lim+ h→0

T hu − u T hu − u = Pj lim+ Σkn=1 Pn h h h→0 h T Pj u − P j u = lim . h h→0+

(3.68)

This yields that Pj u is a solution to Eq.(3.54) with corresponding fj = Pj f . Remark 3.8. If in Theorem 3.12 we assume furthermore that f and u are both almost periodic, then the spectral components uj , j = 1, · · · , k are all almost periodic. This is not the case if neither u, nor f is almost periodic. However, if we have some additional information on the spectral sets Sj , e.g., their countability and the phase space X does not contain c0 , then the almost periodicity of uj are guaranteed. Now we are going to focus our special attention on autonomous equations of the form dx/dt = Ax + f (t),

(3.69)

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where A is the generator of a C0 -semigroup (T (t))t≥0 , and f ∈ BU C(R, X) has precompact range. Below we will use the following notation: σi (A) = {λ ∈ R : iλ ∈ (σ(A) ∩ iR)}. By mild solutions of Eq.(3.69) we will understand in a standard way that they are solutions to Eq.(3.54) with U (t, s) := T (t − s), ∀t ≥ s. As shown below, in this case we can refine the spectral decomposition technique to get stronger assertions which usefulness will be shown in the next subsection when we deal with quasi-periodic solutions. To this purpose, we now prove the following lemma. Lemma 3.16. Let Eq.(3.69) satisfy the above conditions, i.e., A generates a C0 -semigroup and f ∈ BU C(R, X) has precompact range. Moreover, let u be a bounded uniformly continuous mild solution to Eq.(3.69). Then the following assertions hold: (i) (ii)

sp(u) ⊂ σi (A) ∪ sp(f ),

(3.70)

sp(u) ⊃ sp(f ).

(3.71)

Proof. (i) For (3.70) we compute the Carleman transform of u. For Reλ > 0, Z ∞ u ˆ(λ) = e−λt u(t)dt, 0 Z t Z ∞ Z ∞ −λt −λt = e T (t)u(0)dt + e T (tξ)f (ξ)dξ dt 0 0 0 Z t Z ∞ = R(λ, A)u(0) + e−λt T (tξ)f (ξ)dξ dt. (3.72) 0

0

Note that in the same way as in the proof of Lemma 3.14 we can easily show that Z t sp t → T (t − ξ)f (ξ)dξ ⊂ sp(f ). (3.73) 0

Hence, (3.70) is proved. (ii) Note that since for every h > 0 the operator T h is a multiplication by a bounded operator T (h) we have sp(T h u) ⊂ sp(u). Thus, using the argument of the proof of Lemma 3.15 (ii) we have T hu − u sp(f ) = sp(−f ) = sp lim h h→0+ ⊂ sp(u) . This completes the proof of (ii).

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The main result for the autonomous case is the following: Theorem 3.13. Let A generate a C0 -semigroup and f ∈ BU C(R, X) have precompact range. Moreover, let u be a bounded uniformly continuous mild solution to Eq.(3.69). Then the following assertions hold true: (i) If eiσi (A) \σ(f ) is closed, Eq.(3.69) has a bounded uniformly continuous mild solution w such that σ(w) = σ(f ), (ii) If σi (A) is bounded and σi (A)\sp(f ) (3.74) is closed, then Eq.(3.69) has a bounded uniformly continuous mild solution w such that sp(w) = sp(f ). Proof. (i) Note that in this case together with (3.70) the proof of Theorem 3.12 applies. (ii) Under the assumptions there exists a continuous function ψ which belongs to the Schwartz space of all C ∞ -functions on R with each of its derivatives decaying faster than any polynomial such that its Fourier transform ψ˜ has σi (A)\sp(f ) as its support (which is compact in view of the assumptions). Hence, every bounded uniformly continuous function g such that sp(g) ⊂ σi (A) ∪ sp(f ) can be decomposed into the sum of two spectral components as follows: g = g1 + g2 = ψ ∗ g + (g − ψ ∗ g), where g1 = ψ ∗ g, g2 = (g − ψ ∗ g) . Moreover, this decomposition is continuous in the following sense: If g (n) , n = 1, 2, · · · is a sequence in BU C(R, X) with sp(g (n)) ⊂ σi (A) ∪ sp(f ) such that limn g (n) = g in BU C(R, X) , then (n) (n) limn g1 = g1 , limn g2 = g2 . Hence we have in fact proved a version of Theorem 3.11 which allows us to employ the proof of Theorem 3.12 for this assertion (ii). Remark 3.9. (1) In view of the failure of the Spectral Mapping Theorem for general C0 -semigroups the condition in the assertion (i) is a little more general than that formulated in terms of σ(T (1)). (2) If we know beforehand that u is almost periodic, then in the statement of Theorem 3.13 we can claim that the spectral component w is almost periodic.

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Spectral Criteria For Almost Periodic Solutions

This subsection will be devoted to some applications of the spectral decomposition theorem to prove the existence of almost periodic solutions with specific spectral properties. In particular, we will revisit the classical result by Massera on the existence of periodic solutions as well as its extensions. To this end, the following notion will play the key role. Definition 3.14. Let σ(f ) and σΓ (P ) be defined as above. We say that the set σ(f ) and σΓ (P ) satisfy the spectral separation condition if the set σΓ (P )\σ(f ) is closed. Corollary 3.8. Let f be almost periodic, σ(f ) and σΓ (P ) satisfy the spectral separation condition. Moreover, let σ(f ) be countable and X not contain any subspace which is isomorphic to c0 . Then if there exists a bounded uniformly continuous solution u to Eq.(3.54), there exists an almost periodic solution w to Eq.(3.54) such that σ(w) = σ(f ). Proof. We define in this case S1 := σ(f ), S2 := σΓ (P )\σ(f ). Then, by Theorem 3.12 there exists a solution w to Eq.(3.54) such that σ(w) ⊂ σ(f ). Using the estimate (3.64) we have σ(w) = σ(f ). In particular, since σ(w) is countable and X does not contain c0 , w is almost periodic.

Remark 3.10. (1) If σ(f ) is finite, then sp(w) is discrete. Thus, the condition that X does not contain any subspace isomorphic to c0 can be dropped. (2) In the case where σΓ (P ) is countable it is known that with additional ergodic conditions on u the solution u has ”similar spectral properties” as f . However, in many cases it is not expected that the solution u itself has similar spectral properties as f as in the Massera-type problem (see [Massera (68)], [Chow and Hale (20)], [Shin and Naito (96)], [Naito, Minh, R. Miyazaki and Shin (75)] e.g.). (3) In the case where P is compact (or merely σΓ (P ) is finite) the spectral separation condition is always satisfied. Hence, we have a natural extension of a classical result for almost periodic solutions. In this case see also Corollary 3.9 below. (4) We emphasize that the solution w in the statement of Corollary 3.8 is a ”σ(f )-spectral component” of the bounded solution u. This will be helpful to find the Fourier coefficients of w as part of those of u.

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(5) In view of estimate (3.64) w may be seen as a ”minimal” solution in some sense. Corollary 3.9. Let all assumptions of Corollary 3.8 be satisfied. Moreover, let σΓ (P ) be countable. Then if there exists a bounded uniformly continuous solution u to Eq.(3.54), it is almost periodic. Moreover, the following part of the Fourier series of u Z T 1 e−iλξ u(ξ)dξ, (3.75) Σbλ eiλt , bλ = lim T →∞ 2T −T where eiλ ∈ σ(f ), is again the Fourier series of another almost periodic solution to Eq.(3.54). Proof. The assertion that u is almost periodic is standard in view of (3.63) (see Chapter 1). It may be noted that in the case u is almost periodic, the spectral decomposition can be carried out in the function space AP (X) instead of the larger space BU C(R, X). Hence, we can decompose the solution u into the sum of two almost periodic solutions with spectral properties described in Theorem 3.12. Using the definition of Fourier series of almost periodic functions we arrive at the next assertion of the corollary. The next corollary will show the advantage of Theorem 3.13 which allows us to take into account the structure of sp(f ) rather than that of σ(f ). To this end, we introduce the following terminology. A set of reals S is said to have an integer and finite basis if there is a finite subset T ⊂ S such that any element s ∈ S can be represented in the form s = n1 b1 + · · · + nm bm , where nj ∈ Z, j = 1, · · · , m, bj ∈ T, j = 1, · · · , m. If f is quasi-periodic and the set of its Fourier-Bohr exponents is discrete (which coincides with sp(f ) in this case), then the spectrum sp(f ) has an integer and finite basis (see p.48 in [Levitan and Zhikov (58)]). Conversely, if f is almost periodic and sp(f ) has an integer and finite basis, then f is quasi-periodic. We refer the reader to pp. 42-48 in [Levitan and Zhikov (58)] more information on the relation between quasi-periodicity and spectrum, Fourier-Bohr exponents of almost periodic functions. Corollary 3.10. Let all assumptions of the second assertion of Theorem 3.13 be satisfied. Moreover, assume that X does not contain c0 . Then if sp(f ) has an integer and finite basis, Eq.(3.69) has a quasi-periodic mild solution w with sp(w) = sp(f ).

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Proof. Under the corollary’s assumptions the spectrum sp(w) of the solution w, as described in Theorem 3.13, is in particular countable. Hence w is almost periodic. Since sp(w) = sp(f ), sp(w) has an integer and finite basis. Thus w is quasi-periodic. Below we will consider some particular cases Example 3.7. Periodic solutions. If σ(f ) = {1} we are actually concerned with the existence of periodic solutions. Hence, Corollary 3.8 extends the classical result to a large class of evolution equations which has 1 as an isolated point of σΓ (P ). Moreover, Corollary 3.9 provides a way to approximate the periodic solution. In particular, suppose that σΓ (P ) has finitely many elements, then we have the following: Corollary 3.11. Let σΓ (P ) have finitely many elements {µ1 , · · · , µN } and u(·) be a bounded uniformly continuous solution to Eq.(3.54). Then it is of the form u(t) = u0 (t) +

N X

eiλk t uk (t),

(3.76)

k=1

where u0 is a bounded uniformly continuous mild 1-periodic solution to the inhomogeneous equation (3.54), uk , k = 1, · · · , N, are 1-periodic soluPN tions to Eq.(3.54) with f = −iλk uk , respectively, v(t) = k=1 eiλk t uk (t) is a quasi periodic solution to the corresponding homogeneous equation of Eq.(3.54) and λ1 , · · · , λN are such that 0 < λ1 , · · · , < λN < 2π and eiλj = µj , j = 1, · · · , N . Example 3.8. Anti-periodic solutions. An anti-periodic (continuous) function f is defined to be a continuous one which satisfies f (t + ω) = −f (t), ∀t ∈ R and here ω > 0 is given. Thus, f is 2 − ω-periodic. It is known that, the space of anti-periodic functions f with antiperiod ω, which is denoted by AP (ω) , is a subspace of BU C(R, X) with spectrum 2k + 1ω k ∈ Z}. sp(f ) ⊂ { , Without loss of generality we can assume that ω = 1. Obviously, σ(f ) = {−1}, ∀f ∈ AP (ω). In this case the spectral separation condition is nothing but the condition that {−1} is an isolated point of σΓ (P ).

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Example 3.9. Let u be a bounded uniformly continuous solution to Eq.(3.54) with f 2periodic. Let us define f (t) − f (t + 1) f (t) + f (t + 1) F (t) = , G(t) = , ∀t ∈ R. 2 2 Then, it is seen that F is 1-anti-periodic and G is 1-periodic. Applying Theorem 3.12 we see that there exist two solutions to Eq.(3.54) as two components of u which are 1-antiperiodic and 1-periodic with forcing terms F, G, respectively. In particular, the sum of these solutions is a 2-periodic solution of Eq.(3.54) with forcing term f . Example 3.10. Let A be a sectorial operator in a Banach space X and the map t 7→ B(t) ∈ L(Xα , X) be H¨ older continuous and 1-periodic. Then, as shown in Theorem 7.1.3 in [Henry (46)] the equation dx = (−A + B(t))x, (3.77) dt generates a 1-periodic strongly continuous evolutionary process (U (t, s)) t≥s . If, furthermore, A has compact resolvent, then the monodromy operator P of the process is compact. Hence, for every almost periodic function f the sets σ(f ) and σΓ (P ) always satisfy spectral separation condition. In Section 1 we have shown that if σΓ (P ) ∩ σ(f ) = , then there is a unique almost periodic solution xf to the inhomogeneous equation dx = (−A + B(t))x + f (t) (3.78) dt with property that σ(xf ) ⊂ σ(f ) . Now suppose that σΓ (P )∩σ(f ) 6= . By Corollary 3.8, if u is any bounded solution (the uniform continuity follows from the boundedness of such a solution to Eq.(3.78)), then there exists an almost periodic solution w such that σ(w) = σ(f ). We refer the reader to [Henry (46)] and [Pazy (90)] for examples from parabolic differential equations which can be included into the abstract equation (3.78). Example 3.11. Consider the heat equation in materials    vt (t, x) = ∆v(t, x) + f (t, x), t ∈ R, x ∈ Ω   v(t, x) = 0, t ∈ R, x ∈ ∂Ω,

(3.79)

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where Ω ⊂ Rn denotes a bounded domain with smooth boundary ∂Ω. Let X = L2 (Ω), A = ∆ with D(A) = W 2,2 (Ω)∩W01,2 (Ω) . Then A is self-adjoint and negative definite (see e.g. [Pazy (90)]). Hence σ(A) ⊂ (−∞, 0). In particular σi (A) = . Eq.(22) now becomes dv = Av + f. (3.80) dt We assume further that f (t, x) = a(t)g(x) where a is a bounded uniformly continuous real function with sp(a) = Z ∪ πZ, g ∈ L2 (Ω), g 6= 0. It may be seen that σ(f ) = S 1 and sp(f ) has an integer and finite basis. Hence, Theorem 3.12 does not give any information on the existence of a solution w with specific spectral properties. However, in this case Theorem 3.13 applies, namely, if Eq.(3.79) has a bounded solution, then it has a quasi periodic solution with the same spectrum as f .

3.4 3.4.1

Comments and Further Reading Guide Further Reading Guide

The almost periodicity of a bounded solution to the evolution equation u0 (t) = Au(t)+f (t), t ∈ R is an interesting topic in the asymptotic behavior of evolution equations. The method of sums of commuting operators has been widely used in the study of the existence and regularity of solutions on finite intervals. The reader can find more in [Prato and Grisvard (23); Pruss (91)]. There is another approach to the admissibility theory via an operator equation AX − XB = C. The reader can find details of this method in [Vu and Schuler (103); Schweiker (94)]. Recently, the method of Fourier multipliers (see [Weis (104)]) has been used to study the existence and regularity of periodic solutions of inhomogeneous equations in [Arendt and Bu (8); Keyantuo and Lizama (51)], the stability and control in [Latushkin and R¨ abiger (57)]. There are many deep results discussing the conditions for a bounded and uniformly continuous mild solution u of this equation to be almost periodic. These conditions are stated in terms of the countability of imaginary spectrum of A and the phase space not containing c0 . In this direction we refer the reader to [Arendt, Batty, Hieber and Neubrander (7); Levitan and Zhikov (58)] and the references therein for more details. The decomposition method was first studied in [Naito, Minh, R. Miyazaki and Shin (75); Naito, Minh and Shin (76)]. Massera type criteria for the existence of almost periodic solutions with the same structure of spectrum as f are studied in detail for general classes

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of abstract functional differential equations in [Furumochi (35); Murakami, Naito and Minh (71); Minh and Minh (72)]. In [Minh and Minh (72)] the countability of spectrum condition is relaxed considerably. 3.4.2

Comments

The method of evolution semigroups was first used in [Naito and Minh (74)] to study the existence and uniqueness of almost periodic solutions. Subsequently, it was used in [Batty, Hutter and R¨ abiger (14); Murakami, Naito and Minh (71); Naito, Minh, R. Miyazaki and Shin (75); Naito, Minh and Shin (76)] and others. Section 1 is mainly taken from [Naito and Minh (74); Murakami, Naito and Minh (71)]. Section 2 is the main part of [Murakami, Naito and Minh (71)]. Section 3 with the decomposition method is taken from [Naito, Minh and Shin (76)]. The upper spectral estimate (3.63) was first found by Zhikov (see [Levitan and Zhikov (58)]) for autonomous equations. It was extended to periodic equations in [Batty, Hutter and R¨ abiger (14)]. The lower spectral estimate (3.64) was first found in [Naito, Minh and Shin (76)].

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Chapter 4

Almost Automorphic Solutions

In this chapter we will study conditions under which an evolution equation has an almost automorphic (classical/mild) solution. The concept of almost automorphic functions was introduced by S. Bochner three decades ago. However, only recently the interest in almost automorphic solutions of evolution equations has been resurgent. We begin this chapter with some fundamental results on almost automorphic solutions of evolution equations in Section 4.1. In the next section we will present a new method using invariant subspaces. An existence result based on Fixed Point Theorems for semilinear equations will be given in Section 4.3. One difficulty we have to overcome in this section is due to the fact that almost automorphic functions are not necessarily uniformly continuous. We deal with this situation in Sections 4.4 and 4.5 by using the method of sums of commuting operators and the notion of uniform spectrum recently introduced in [Diagana, N’Gu´er´ekata and Minh (28)]. In Section 4.5, we are concerned with a second-order linear evolution equation.

4.1

The Inhomogeneous Linear Equation

In this section we will consider in a (real or complex) Banach space X the differential equation x(t) ˙ = Ax(t) + f (t),

t∈R

(4.1)

where f ∈ AA(X), the Banach space of all almost automorphic functions R 7→ X (see Section 1.3.7) and A is a (generally unbounded, unless otherwise stated) linear operator with domain D(A) ⊂ X. Our concern is to look at 121

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conditions under which (classical or mild) bounded solutions of (4.1) are also in AA(X). This problem was initially raised and solved by Bohr and Neugebauer in the case where f ∈ AP (X) and X is a finite dimensional space. The almost automorphic version of this result is well-known. We begin our study with the following result Theorem 4.1. If A = λ is a complex number and f ∈ AA(Cn ), then the solution x(t) of (4.1) is in AA(Cn ). Proof. case,

We first consider the case where λ = iθ, where θ ∈ R. In this x(t) = eiθt x(0) +

Z

t

eiθ(t−ξ) f (ξ)dξ,

0

∀t ∈ R.

Therefore, from theR boundedness of x(·) it follows the boundedness of t the function t 7→ 0 eiθ(t−ξ) f (ξ)dξ. Consequently, the function t 7→ R t iθ(−ξ) f (ξ)dξ is bounded. Observe that the function ξ 7→ eiθ(−ξ) f (ξ)dξ 0 e Rt is almost automorphic, and so the integral function t 7→ 0 eiθ(−ξ) f (ξ)dξ is almost automorphic. Finally, we get the almost automorphy of the bounded solution x(·). If <e λ 6= 0, then x(t) is equal to x1 (t) if <e λ > 0, and x2 (t) if <e λ < 0, where Z ∞ eλ(t−s) f (s)ds t ∈ R, x1 (t) = − t

x2 (t) =

Z

t −∞

eλ(t−s) f (s)ds t ∈ R

Let us prove that x1 (t) ∈ AA(Cn ). The proof of x2 (t) ∈ AA(Cn ) is similar. Use the change of variable σ = t − s to obtain Z 0 x1 (t) = − eλσ f (t − σ)dσ t ∈ R −∞

0

Let now (σn ) be an arbitrary sequence of real numbers. Since f ∈ 0 AA(Cn ), there exists a subsequence (σn ) of (σn ) such that g(t) = lim f (t + σn ) n→∞

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is well defined for each t ∈ R, and f (t) = lim g(t − σn ) n→∞

for each t ∈ R. Now fix t ∈ R. Then we get lim f (t − σ + σn ) = g(t − σ)

n→∞

for each given σ ∈ R. Clearly x1 (t + σn ) = −

Z

0

−∞

eλσ f (t − σ + σn ) dσ

We observe also that keλσ f (t − σ + σn )| ≤ e(<e λ)σ sup kf (t)k t∈R

and Z

0 −∞

e(<eλ)σ sup kf (t)k dσ = t∈R

1 sup kf (t)k < ∞ <eλ t∈R

We know also the g is a bounded and measurable function. Using now the Lebesgue’s dominated convergence theorem, we obtain Z 0 lim x1 (t − σn ) = − eλσ g(t − σ) dσ = y1 (t) n→∞

−∞

for each t ∈ R. We can apply the same reasoning to y1 (t) to obtain lim y1 (t − σn ) = x1 (t)

n→∞

for each t ∈ R, which proves that x1 (t) ∈ AA(Cn ) and completes the proof.

Now we have: Theorem 4.2. ([N’Gu´er´ekata (79)][Remark 4.2.2]) If A is a linear operator Rn 7→ Rn , then every bounded solution of (4.1) is in AA(Rn ). Proof. First, we note that by Floquet Theory of periodic ordinary differential equations, without loss of generality we may assume that A is independent of t. Next we will show that the problem can be reduced to the onedimensional case. In fact, if A is independent of t, by a change of variable

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if necessary, we may assume that A is of Jordan normal form. In this direction we can go further with assumption that A has only one Jordan box. That is, we have to prove the theorem for equations of the form      f (t)  x˙ 1 (t) 1 λ 1 0 ... 0  x˙ 2 (t)    f2 (t)   0 λ 1 . . . 0    +  . =  . .  ..  . . . . . . . . . . . .   ..  0 0 0 ... λ x˙ n (t) fn (t) Let us consider the last equation involving xn (t). We have x˙ n (t) = λxn (t) + fn (t),

t ∈ R, x(t) ∈ C.

(4.2)

If <λ 6= 0, then we can easily check that either Z t y(t) = eλ(t−ξ) fn (ξ)dξ, (<λ < 0) −∞

or

z(t) =

Z

∞

eλ(t−ξ) fn (ξ)dξ,

(<λ > 0)

t

is a unique bounded solution of Eq. (4.2) Moreover, by Theorem 4.1, in both cases, y(t) and z(t) are in AA(C). Hence, xn is in AA(C). If <λ = 0, then λ = iη for η ∈ R. By assumption, there is a constant c such that the function Z t xn (t) := ceiηt + eiη(t−ξ) f (ξ)dξ, 0

Rt is bounded on R. This yields the boundedness of 0 e−iηξ f (ξ)dξ on R. Rt Hence, 0 e−iηξ f (ξ)dξ is in AA(X). Finally, this yields that xn is in AA(X). Let us consider next the equation involving xn−1 and xn . Since xn is in AA(X), by repeating the above argument we can show that xn−1 is also in AA(X). Continuing this process, we can show that all xk (·) are in AA(X). The proof is completed. Suppose now f ∈ C(R, X) ∩ L1 (R+ , X) and A is the infinitesimal generator of a C0 -group (T (t))t∈R . Consider the function x ∈ C(R, X) defined by Z t T (t − s)f (s)ds, t ∈ R x(t) = T (t)x(0) + 0

Let us assume that T (t)y ∈ AA(X) for every y ∈ X, which implies that T (−t)y ∈ AA(X) for every y ∈ X.

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Now we write x(t) = u(t) + v(t),

t∈R

where u(t) = T (t)x(0) +

Z

∞ 0

T (t − s)f (s)ds

and v(t) = −

Z

t

∞

T (t − s)f (s)ds.

It is clear that u ∈ AA(X) by Theorem 1.19 i) and v ∈ C(R+ , X) with limt→∞ kv(t)k = 0. Finally x(t)|t∈R+ is an asymptotically almost automorphic function. We continue our study of (4.1) with the following result ([N’Gu´er´ekata (82)]). Theorem 4.3. Let A be the infinitesimal generator of a C0 -group of bounded linear operators (T (t))t∈R . Assume that there exists a subspace X1 of D(A) with dim X1 < ∞ such that: i) AX1 ⊂ X1 ii) (T (t) − I)f (s) ∈ X1 , ∀s, t ∈ R iii) A commutes with the projection P : X 7→ X1 , on D(A). Then every solution x(t) of (4.1) with x(0) ∈ X1 and precompact range Rx = {x(t)/t ∈ R} is in AA(X). Proof. Basically the proof uses the so-called method of decomposition of the space in order to apply a result of the Bohr-Neugebauer type (Theorem 4.1). We let P : X 7→ X1 , be the natural projection on X1 and consider the associated decomposition X = X1 ⊕ kerP where kerP = {x ∈ X/P x = 0} is the kernel of P . Let Q = I − P ; then QX1 = {0} and Q2 = I. We recall also that both P and Q are bounded linear operators. Now if x(t) is a solution of (4.1) as given in the Theorem, then we can write x(t) = x1 (t) + x2 (t),

t∈R

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where x1 (t) = P x(t) ∈ X1 and x2 (t) = Qx(t) ∈ kerP for each t ∈ R. Since Rx is precompact, it is also clear that both Rx1 and Rx2 are precompact. We have x(t) = T (t)x(0) +

Z

t 0

T (t − s)f (s)ds =

Z t Z t T (t)x(0) + f (s)ds + (T (t − s) − I)f (s)ds 0 0 Rt Note that 0 (T (t − s) − I)f (s)ds ∈ X1 , by ii) in the Theorem, hence Rt Q 0 (T (t − s) − I)f (s)ds = 0. Consequently Z t Z t Qf (s)ds f (s)ds = QT (t)x(0) + x2 (t) = Qx(t) = QT (t)x(0) + Q 0

0

Now

x˙ 2 (t) = QAT (t)x(0) + Qf (t) = QT (t)Ax(0) + Qf (t) = Qf (t) Since Ax(0) ∈ X1 , that yields T (t)Ax(0) ∈ X1 and consequently QT (t)Ax(0) = 0. By Exercise 4.1 below, Qf (t) is almost automorphic; so is x˙ 2 (t). Now since Rx2 is precompact, we deduce that x2 is in AA(X). Now it suffices to prove that x1 (t) is almost automorphic in order to reach the conclusion based on Theorem 1.19 i). We rewrite (4.1) as follows x(t) ˙ = Ax1 (t) + Ax2 (t) + P f (t) + Qf (t) and we apply P to both sides of the previous equation to get x˙ 1 (t) = P Ax1 (t) + P Ax2 (t) + P 2 f (t) + P Qf (t) = P Ax1 (t) + AP x2 (t) + P 2 f (t) + P Qf (t) = P Ax1 (t) + g(t) where g(t) = P 2 f (t) + P Qf (t) is almost automorphic. This last equation holds true in the finite dimensional space X1 . It is of the Borh-Neugebauer type (Theorem 4.1). So its solution x1 (t) is almost automorphic. The proof is achieved. Exercise 18. Let f ∈ AA(X) and A ∈ B(X), the space of all bounded linear operators on X. Show that Af ∈ AA(X). Exercise 19. Let H be a Hilbert space and A : H 7→ H a linear operator such that Aφn = λn φn n = 1, 2, 3, ... where {φ1 , φ2 , ...} is an orthonormal base for H and |<eλn | ≤ M for all n = 1, 2, 3, .... Find the subspace X1 of H such that AX1 ⊂ X1 , and A commutes with the projection P : H 7→ X1 .

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Method of Invariant Subspaces and Almost Automorphic Solutions of Second-Order Differential Equations

In this section we are concerned with the almost automorphic solutions to the homogeneous second-order hyperbolic differential equation of the form d2 d u(s) + 2B u(s) + A u(s) = 0, ds2 ds and the associated nonhomogeneous differential equation

(4.3)

d2 d u(s) + 2B u(s) + A u(s) = f (s), (4.4) 2 ds ds where A, B are densely defined closed linear operators acting in a Hilbert space H and f ∈ AA(H). We use invariant subspaces theory (see Appendix 6.4) to show that under appropriate assumptions, every solution to the equations (4.2) and/or (4.3) is an almost automorphic vector-valued function. The idea of using the method of invariant subspaces to study the existence of almost automorphic solutions is recent and due to Diagana and N’Gu´er´ekata. Let us indicate that the invariant subspaces method works smoothly in the framework of abstract differential equations involving the algebraic sum of unbounded linear operators. d u(s), the problem (4.2)-(4.3) can be rewritten in Now setting v(s) = ds H × H of the form d U(s) = (A + B) U(s), (4.5) ds and d U(s) = (A + B) U(s) + F (s), (4.6) ds where U(s) = (u(s), v(s)), F (s) = (0, f (s)) and A, B are the operator matrices of the form O I O O A = and B = , −A O O −2B on H × H with D(A) = D(A) × H, D(B) = H × D(B), and O, I denote the zero and identity operators on H, respectively. Since (4.2)-(4.3) is equivalent to (4.4)-(4.5), instead of studying (4.2)(4.3), we will focus on the characterization of almost automorphic solutions to (4.4)-(4.5).

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Throughout the section, H, D(C), R(C) and N (C), denote a Hilbert space, the domain, the range and the kernel of the linear operator C, respectively. Let A and B be densely defined closed unbounded linear operators on H. Recall that their algebraic sum is defined by D(A + B) = D(A) ∩ D(B) and (A + B)x = Ax + Bx, ∀x ∈ D(A) ∩ D(B). Since both A and B are densely defined, then the algebraic sum of A and B, S = A + B is also a densely defined operator and D(A + B) = D(A) × D(B), and (A + B)U = AU + BU. Throughout the section, A and B will play similar roles. Setting our main result, we make the following assumptions: The operators A and B are infinitesimal generators of C0 -groups of bounded operators (T (t))s∈R , (R(t))s∈R , respectively, such that (i) T (s)U : s 7→ T (s)U is almost automorphic for each U ∈ H × H, R(s)V : s 7→ R(s)V is almost automorphic for each V ∈ H × H, respectively. (ii) there exists S ⊂ H × H, a closed subspace that reduces both A and B. We denote by PS and QS = (I × I − PS ) = P[H×H] S , the orthogonal projections onto S and [H × H] S, respectively. (iii) R(A) ⊂ R(PS ) = N (QS ) (iv) R(B) ⊂ R(QS ) = N (PS ) Remark 4.1. 1. Recall that if A, B generate C0 -groups, then their sum A + B need not be a generator of a C0 -group . 2. The assumption (ii) implies that both S and [H×H S] are invariant for the algebraic sum A + B (note that it is well-defined as stated above). Now we state and prove: Theorem 4.4. Under assumptions (i)-(ii)-(iii)-(iv), every solution to the differential equation (4.4) is almost automorphic. Proof. Let X(s) be a solution to (4.4). Clearly X(s) ∈ D(A) ∩ D(B) = D(A)×D(B) (notice that the algebraic sum S = A+B exists since D(S) = H × H).

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Now decompose X(s) as follows X(s) = PS X(s) + (I × I − PS )X(s), where PS X(s) ∈ R(PS ) = N (QS ), and QS X(s) ∈ N (PS ) = R(QS ). We have d d (PS X(s)) = PS X(s) ds ds = PS AX(s) + PS BX(s)

= APS X(s) + PS BX(s) (according to(ii)) = APS X(s) (according to(iv))

From

d (PS X(s)) = APS X(s), it follows that ds PS X(s) = T (t)PS X(0).

Now according to (i), the vector-valued function s 7→ PS X(s) = T (t)PS X(0) is almost automorphic. In the same way, since [H × H] ([H × H] S) = S. It follows that the closed subspace S reduces A and B if and only if [H × H] S does. In other words, [H × H] S reduces A and B. That is, a similar remark as Remark 6.1 in Appendix holds when S is replaced by [H × H] S. Thus, we have d d (QS X(s)) = QS X(s) ds ds = QS AX(s) + QS BX(s)

= QS AX(s) + BQS X(s) (according to(ii)) = BQS X(s) (according to(iii))

d (QS X(s)) = BQS X(s), it follows that s 7→ dt QS X(s) = R(s)QS X(0) is almost automorphic (according to (i)). Therefore X(s) = PS X(s) + QS X(s) is also almost automorphic as the sum of almost automorphic vector-valued functions. From the equation

Corollary 4.1. Let B : H 7→ H be a bounded linear operator in the Hilbert space H. Assume that assumptions (i)-(ii)-(iii)-(iv) hold true. Then every solution to the equation (4.5) is almost automorphic. Proof. This an immediate consequence of Theorem 4.5 to the case where B is a bounded linear operator, it is straightforward.

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Consider now the nonhomogeneous equation (4.5). Assume that the vector valued function f : R 7→ H is almost automorphic. In fact, this implies that F : s 7→ (0, f (s)) is in AA(H × H). We have Theorem 4.5. Under assume in addition R t assumption (i)-(ii)-(iii)-(iv), Rt that f ∈ AA(H) and 0 T (−s)f (s)ds and 0 R(−s)f (s)ds are bounded over R. Then every solution to the equation (4.5) is almost automorphic. Proof. Let X(s) be a solution to (4.5). As in the proof of Theorem 4.5, the solution X(s) ∈ D(A) ∩ D(B). Now express X(s) as X(s) = PS X(s) + QS X(s), where PS , QS = (I × I − PS ) = P[H×H] S are the orthogonal projections defined above. We have d d (PS X(s)) = PS X(s) ds ds = PS AX(s) + PS BX(s) + PS F (s)

= APS X(s) + PS BX(s) + PS F (s) (according to(ii)) = APS X(s) + PS F (s) (according to(iv))

From

d (PS X(s)) = APS X(s) + PS F (s); it follows that ds Z s PS X(s) = T (s)PS X(0) + T (s − σ)PS F (σ)dσ. 0

Rs

Set G(s) = 0 T (s − σ)PS F (σ)dσ . First observe that σ 7→ RT (−σ) PS F (σ) is almost automorphic. Mores over the function x(s) =: 0 T (−σ)PS F (σ)dσ is bounded (as it can be easily proved), thus it is almost automorphic. Now T (s) x(s) = G(s) is almost automorphic. According to assumption (i), the vector-valued function s 7→ PS X(s) = T (s)PS X(0) is almost automorphic. Therefore s 7→ PS X(s) is almost automorphic as the sum of almost automorphic vector-valued functions. In the same way, it is not hard to see that d (QS X(s)) = BQS X(s) + QS F (s), ds

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and that QS X(s) can be expressed as QS X(s) = R(s)QS X(0) +

Z

s 0

R(s − σ)QS F (σ)dσ.

Using similar arguments as above, it can be shown that s 7→ QS X(s) is almost automorphic. Therefore X(s) = PS X(s)+QS X(s) is also an almost automorphic vector-valued function. Remark 4.2. Let us notice that the previous results (Theorem 4.5 and Theorem 4.6) still hold in the case where A, B : H × H 7→ H × H are bounded linear operator matrices on H × H. In such a case, the similar assumptions are required, that is, (i)-(ii)-(iii) and (iv). 4.3

Existence of Almost Automorphic Solutions to Semilinear Differential Equations

This section is concerned with the differential equation in a Banach space X: x0 (t) = Ax(t) + f (t, x(t)), t ∈ R

(4.7)

where A is the infinitesimal generator of an exponentially stable C0 semigroup (T (t))t≥0 ; that is, there exist K > 0, ω < 0 such that kT (t)k ≤ Keωt , ∀t ≥ 0. We assume that f : R×X 7→ X satisfies a Lipschitz condition in x uniformly in t, that is, kf (t, x) − f (t, y)k ≤ Lkx − yk ∀t ∈ R, x, y ∈ X. We like to establish existence and uniqueness of almost automorphic mild solutions to the Eq. (4.6). We first prove the existence of almost automorphic mild solution of the differential equation x0 (t) = Ax(t) + f (t), t ∈ R,

(4.8)

where f is almost automorphic. Theorem 4.6. Let f ∈ AA(X). Then equation (4.7) has a unique almost automorphic mild solution.

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Proof. We first prove existence of an almost automorhic solution. It is well-known that Eq. (4.7) possesses a mild solution of the form Z t x(t) = T (t − a)x(a) + T (t − s)f (s)ds, f or all a ∈ R, t ≥ a. a

It remains to prove that it is almost automorphic. Rt First, we consider the function u(t) := −∞ T (t − s)f (s)ds, defined as Z t Z t T (t − s)f (s)ds = lim T (t − s)f (s)ds. r→−∞

−∞

r

Rt

Clearly for each r < t, the integral r T (t − s)f (s)ds exists. Moreover Z t K kf k∞ , ∀r < t. k T (t − s)f (s)dsk ≤ |ω| r Rt which shows −∞ T (t − s)f (s)ds is absolutely convergent. Now let (s0n ) be an arbitrary sequence of real numbers. Since f ∈ AA(X), there exists a subsequence (sn ) of (s0n ) such that g(t) = lim f (t + sn ) n→+∞

is well-defined for each t ∈ R and f (t) = lim g(t − sn ) n→+∞

for each t ∈ R Now consider u(t + sn ) = = =

t+sn

Z

−∞ Z t

−∞ Z t −∞

T (t + sn − s)f (s)ds T (t − σ)f (σ + sn )dσ T (t − σ)fn (σ)dσ,

where fn (σ) = f (σ + sn ), n = 1, 2, ... We also have ku(t + sn )k ≤

K kf k∞ , ∀n = 1, 2, ... |ω|

and by continuity of the semigroup, T (t − σ)fn (σ) 7→ T (t − σ)g(σ), as n → ∞ for each σ ∈ R fixed and any t ≥ σ.

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Rt If we let v(t) = −∞ T (t − s)g(s)ds, we observe that the integral is absolutely convergent for each t. So, by the Lebesgue’s dominated convergent theorem, u(t + sn ) 7→ v(t), as n → ∞

for each t ∈ R. We can show in a similar way that

v(t − sn ) 7→ u(t) as n → ∞

for each t ∈ R. This Rshows that u ∈ AA(X). Ra a Now let u(a) = −∞ T (a − s)f (s)ds. So T (t − a)u(a) = −∞ T (t − s)f (s)ds. If t ≥ a, then Z t Z t Z a T (t − s)f (s)ds = T (t − s)f (s)ds − T (t − s)f (s)ds a

−∞

−∞

= u(t) − T (t − a)u(a). Rt so that, u(t) = T (t − a)u(a) + a T (t − s)f (s)ds. If we fix x(a) = u(a), then x(t) = u(t), that is x ∈ AA(X). We finally prove the uniqueness of the almost aumorphic solution. Assume x and y are two such solutions and we let z = x − y. Then z ∈ AA(X) and satisfies the equation z 0 (t) = Az(t), t ∈ R.

Note that z is bounded and satisfies also the equation z(t) = T (t − s)z(s) ∀t, s ∈ R, t ≥ s We also have the inequality kz(t)k ≤ Keω(t−s) .

Take a sequence of real numbers (sn ) such that sn → −∞. For any fixed t ∈ R, we then can find a subsequence (snk ) of (sn ) with snk < t for all k = 1, 2, .... Using the fact that ω < 0, we obtain z = 0. This shows uniqueness of the solution and ends the proof. Theorem 4.7. Assume that f : R × X 7→ X satisfies a Lipschitz condition in x uniformly in t, that is, kf (t, x) − f (t, y)k ≤ Lkx − yk, ∀x, y ∈ X,

|ω| . Let also f be almost automorphic in t for each x ∈ X. K Then equation (4.6) has a unique almost automorphic mild solution. where L <

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Proof. Let x be a mild solution of Eq. (4.6). It is continuous and satisfies the integral equation Z t x(t) = T (t − a)x(a) + T (t − s)f (s, x(s))ds, ∀a ∈ R, ∀t ≥ a. a Rt Consider a T (t − s)f (s, x(s))ds and the nonlinear operator G : AA(X) 7→ AA(X) given by Z t (Gφ)(t) := T (t − s)f (s, φ(s))ds. −∞

In view of Theorem 2.2.6 in [N’Gu´er´ekata (79)], if φ ∈ AA(X), then Gφ ∈ AA(X), so G is well-defined. Now for φ1 , φ2 ∈ AA(X), we have: Z t kGφ1 − Gφ2 k∞ = sup k T (t − s){f (s, φ1 (s)) − f (s, φ2 (s))}dsk t∈R

≤ sup t∈R

Z

−∞ t

−∞

kT (t − s)kB(X) Lkφ1 (s) − φ2 (s)kds

≤ Lkφ1 − φ2 k∞ . sup t∈R

≤ Lkφ1 − φ2 k∞ . sup t∈R

= So

LK kφ1 − φ2 k∞ . |ω|

Z

t

−∞ Z t

kT (t − s)kB(X) ds Keω(t−s) ds

−∞

LK kφ1 − φ2 k∞ , |ω| LK which proves that G is continuous. And since < 1, then G is a con|ω| traction. R t So there exists a unique u ∈ AA(X), such that Gu = u, that is u(t) = −∞ T (t − s)f R a(s, u(s))ds. If we let u(a) = −∞ T (a − s)f (s, u(s))ds, then Z a T (t − a)u(a) = T (t − s)f (s, u(s))ds. kGφ1 − Gφ2 k∞ ≤

But for t ≥ a, Z t a

−∞

Z t T (t − s)f (s, u(s))ds = T (t − s)f (s, u(s))ds −∞ Z a − T (t − s)f (s, u(s))ds −∞

= u(t) − T (t − a)u(a). Rt So u(t) = T (t − a)u(a) + −∞ T (t − s)f (s, u(s))ds is a mild solution of equation (4.6) and u ∈ AA(X). The proof is now complete.

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4.4

135

Method of Sums of Commuting Operators and Almost Automorphic Functions

In this section we will extend the method of sums of commuting operators in the previous chapter to almost automorphic solutions of inhomogeneous linear evolution equations of the form du = Au + f (t), (4.9) dt where A is an (unbounded) linear operator which generates a holomorphic semigroup of linear operators on a Banach space X and f ∈ AA(X). The main difficulty that arises here is concerned with the non-uniform continuity of almost automorphic functions. This implies the non-strong continuity of translation semigroup in the functions space AA(X). Therefore, many elegant proofs using semigroup theory fail. To overcome this difficulty it is appropriate to use the concept of uniform spectrum defined in Chapter 1. For any closed subset Λ ⊂ R we denote AAΛ (X) := {u ∈ AA(X) : spu (u) ⊂ Λ}. By the basic properties of uniform spectra of functions, AAΛ (X) is a closed subspace of BC(R, X). Below we denote DΛ the part of the differential operator d/dt in AAΛ (X). We have the following: Lemma 4.1. Under the above notations and assumptions we have σ(DΛ ) = iΛ.

(4.10)

Proof. The proof can be taken from the one of Lemma 1.2. The details are left to the reader. As a standing assumption in the remaining part of the section we always assume that A is the infinitesimal generator of an analytic semigroup of linear operators on X. Let Λ be a closed subset of R. We first consider the operator AΛ of multiplication by A and the differential operator d/dt on the function space AAΛ (X). By definition the operator AΛ of multiplication by A is defined on D(AΛ ) := {g ∈ AAΛ (X) : g(t) ∈ D(A) ∀t ∈ R, Ag(·) ∈ AAΛ (X)}, and Ag := Ag(·) for all g ∈ D(AΛ ).

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Lemma 4.2. Assume that Λ ⊂ R is closed. Then the operator AΛ of multiplication by A in AAΛ (X) is the infinitesimal generator of an analytic C0 -semigroup on AAΛ (X). Proof. We will prove that AΛ is a sectorial operator on AAΛ (X). In fact, first we check that AΛ is densely defined. Consider the semigroup TΛ (t) of operators of multiplication by T (t) on AAΛ (X). We now show that it is strongly continuous. Indeed, suppose that g ∈ AAΛ (X), since R(g) is relatively compact we see that the map [0, 1] × R(g) 3 (t, x) 7→ T (t)x ∈ X is uniformly continuous. Hence, sup kT (t)g(s) − g(s)k → 0 s∈R

as t → 0, i.e., the TΛ (t) is strongly continuous. By definition, g ∈ D(AΛ ) if and only if g(s) ∈ D(A), ∀s ∈ R and Ag(·) ∈ AAΛ (X). Thus, Z 1 t T (t)g(s) − g(s) T (ξ)Ag(s)dξ, ∀t ≥ 0, s ∈ R. = t t 0 Therefore, lim+ sup k

t→0

s∈R

T (t)g(s) − g(s) 1 − t t

Z

t

T (ξ)Ag(s)dξk = 0, 0

i.e., g is in D(G), where G is the generator of TΛ (t) and AΛ g = Gg. Conversely, we can easily show that G ⊂ AΛ . Now it suffices to prove that σ(AΛ ) ⊂ σ(A) to claim that AΛ is a sectorial operator. In fact, let µ ∈ ρ(A). To prove that µ ∈ ρ(AΛ ) we show that for each h ∈ AAΛ (X) the equation µg − AΛ g = h has a unique solution in AAΛ (X). But this follows from the fact that (µ − AΛ )−1 h(·) ∈ AAΛ (X) and that the equation µx − Ax = y has a unique solution x in X for any y ∈ X.

Theorem 4.8. Let A be the generator of an analytic semigroup. Then the operator AΛ of multiplication by A and the differential operator DΛ on AAΛ (X) are commuting and satisfy condition P (for the definition see the Appendix).

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Proof. By the above lemma the operator AΛ is sectorial. It suffices to show that it commutes with the differential operator DΛ . In fact, since 1 ∈ ρ(DΛ ) we will prove that R(1, DΛ )R(ω, AΛ ) = R(ω, AΛ )R(1, DΛ ),

(4.11)

for sufficiently large real ω. Since AΛ generates the semigroup TΛ (t), using well-known facts from the semigroup theory the above indentity for sufficiently large ω is equivalent to the following Z ∞ Z ∞ e−ωt TΛ (t)dtR(1, DΛ ). (4.12) e−ωt TΛ (t)dt = R(1, DΛ ) 0

0

In turn, (4.12) follows from the following

R(1, DΛ )TΛ (τ ) = TΛ (τ )R(1, DΛ ),

which is obvious.

∀τ ≥ 0.

(4.13)

So, by the spectral properties of sums of commuting operators, we have Corollary 4.2. If σ(A) ∩ iΛ = , then for every f ∈ AAΛ (X) there exists a unique u ∈ AAΛ (X) such that AΛ + DΛ u = f.

Proof. Since AΛ and DΛ commute and satisfy Condition P, the sum AΛ + DΛ is closable (denote its closure by AΛ + DΛ ). From σ(A) ∩ iΛ = and Theorem 1.11 in Appendix, it turns out that 0 ∈ ρ(AΛ + DΛ ). Therefore for every f ∈ AAΛ (X) there exists a unique u ∈ D(AΛ + DΛ ) such that AΛ + DΛ u = f.

Now our remaining task is just to explain what the above closure means. More precisely, we will relate it with the notion of mild solutions to evolution equations. Lemma 4.3. Let u, f ∈ AA(X). If u ∈ D(AΛ + DΛ ) and AΛ + DΛ u = f , then u is a mild solution of Eq. (4.11). Proof. This lemma follows immediately from the following: For every u ∈ AA(X) we say that it belongs to D(L) of an operator L acting on AA(X) if there is a function f ∈ AA(X) such that Z t u(t) = T (t − s)u(s) + T (t − ξ)f (ξ)dξ, ∀t ≥ s, t, s ∈ R. (4.14) s

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By a similar argument as in the proof of Lemma 3.1 in [Murakami, Naito and Minh (71)] we can prove that L is a closed single-valued linear operator acting on AA(X) which is an extension of AΛ + DΛ . Thus, L is an extension of AΛ + DΛ . This yields that u is a mild solution of Eq. (4.11). As an immediate consequence of the above argument we have: Theorem 4.9. Let A be the generator of an analytic semigroup and let Λ be a closed subset of R. Then it is necessary and sufficient for each f ∈ AAΛ (X) there exists a unique mild solution u ∈ AAΛ (X) to Eq. (4.11) that the condition σ(A) ∩ iΛ = holds. Proof. The sufficiency follows from the above argument. The necessity can be shown as follows: For every ξ ∈ Λ, obviously that the function h : R 3 t 7→ aeiξt is in AAΛ (X), where a ∈ X is any given element. By assumption, there is a unique g ∈ D(AΛ ) such that iξg(t) − Ag(t) = h(t) for all t ∈ R. Following the argument in the proof of Theorem 3.4 one can easily show that g(t) is of the form beiξt . Hence, b is the unique solution of the equation iξb − Ab = a. That is iξ 6∈ σ(AΛ ), so iΛ ∩ σ(AΛ ) = . Corollary 4.3. Let A be the generator of an analytic semigroup such that σ(A) ∩ i spu (f ) = . Then Eq. (4.11) has a unique almost automorphic mild solution w such that spu (w) ⊂ spu (f ). Proof.

Set Λ = spu (f ). Then by the above argument we get the theorem.

Remark 4.3. We notice that all results stated above for almost automorphic solutions hold true for compact almost automorphic solutions if the assumption on the almost automorphy of f is replaced by the compact almost automorphy of f . Details of the proofs are left to the reader.

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139

Almost Automorphic Solutions of Second Order Evolution Equations

In this section we deal with the existence of almost automorphic mild solutions to second order evolution equations of the form d2 u = Au + f (t), dt2

(4.15)

where A is an (unbounded) linear operator which generates a holomorphic semigroup of linear operators on a Banach space X and f ∈ AA(X). We will use the method of sums of commuting operators, a method due to Murakami, Naito and Minh [Murakami, Naito and Minh (71)], to study the existence of almost periodic solutions is . This method works well with the problem of finding almost automorphic solutions of first order evolution equations as shown in the previous section 4.4. We will show that the method of sums is still useful for second order evolution equations. Our main results are Theorems 4.9, 4.11. Notice that in [Schweiker (94)], [Schuler and Vu (95)], similar results for f ∈ BU C(R, X) were proved using an operator equation. This method that needs the uniform continuity of f is inapplicable to the problem we are considering due to the fact that an almost automorphic function may not be uniformly continuous. Notation. Throughout the section, R, C, X stand for the sets of all real, all complex numbers and a complex Banach space, respectively; L(X), BC(R, X), BU C(R, X) denote the spaces of all linear bounded operators on X, all X-valued bounded continuous functions, all X-valued bounded uniformly continuous with sup-norm, respectively. The translation group in BC(R, X) is denoted by (S(t))t∈R which is strongly continuous in BU C(R, X) whose infinitesimal generator is the differential operator d/dt. For a linear operator A, we denote by D(A), σ(A) and ρ(A) the domain, spectrum and resolvent set of A, respectively. If Y is a metric space and ¯ denotes its closure in Y . In this section by the B is a subset of Y , then B notion of sectorial operators means the one defined in [Pazy (90)]. The notion of closure of an operator is referred to the one defined in [Davies (27)].

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Mild Solutions of Inhomogeneous Second Order Equations

4.5.1.1 Mild Solutions Let A be any closed linear operator on a Banach space X. We now define the concept of mild solutions on R to Eq. (4.15). Definition 4.1. (See p. 374 in [Arendt and Batty (5)] ) A continuous X-valued function u is called a mild solution on R of Eq. (4.15) if Z t (t − s)u(s)ds ∈ D(A) ∀t ∈ R (4.16) 0

and

u(t) = x + ty + A

Z

t 0

(t − s)u(s)ds +

Z

t 0

(t − s)f (s)ds

(t ∈ R)(4.17)

for some fixed x, y ∈ X. If u ∈ C 2 (R, X), u(t) ∈ D(A), ∀t ∈ R and Eq.(4.15) holds for all t ∈ R we say that u is a classical solution to Eq.(4.15). It is easily seen that if u ∈ C(R, X) is a classical solution of Eq. (4.15), then it is a mild solution of Eq. (4.15). 4.5.1.2 Mild Solutions and Weak solutions Definition 4.2. A function u ∈ BC(R, X) is said to be a weak solution to Eq. (4.15) if there is a sequence of fn ∈ BC(R, X) and a sequence of classical solutions un ∈ BC(R, X) of Eq. (4.15) with f replaced by fn such that fn → f and un → u in the sup-norm topology of BC(R, X). Definition 4.3. We define an operator L on BC(R, X) with domain D(L) consisting of all u ∈ BC(R, X) such that there exists at least a function f ∈ BC(R, X) for which u is a mild solution of Eq. (4.15), that is, (4.16) and (4.17) hold. Lemma 4.4. If A is a closed linear operator, then L is a single-valued closed linear operator BC(R, X) ⊃ D(L) → BC(R, X). Proof. First we show that L is a single-valued linear operator. For this it suffices to show that if u(t) ≡ 0 is a mild solution of Eq. (4.15), then f (t) ≡ 0. In fact, by taking t = 0 we can see that in Eq. (4.17), x = 0. Hence, we have Z t 0 = ty + (t − s)f (s)ds (t ∈ R). (4.18) 0

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Differentiating (4.18) twice, we have f (t) = 0 for all t ∈ R. Therefore, the operator L is a single-valued linear operator. Next, we show its closedness. Let un be in D(L) that are mild solutions of Eq. (4.15) with f replaced by fn ∈ BC(R, X) such that un → u ∈ BC(R, X) and fn → f ∈ BC(R, X). We have to prove that u is a mild solution to Eq. (4.15). Indeed, by our assumption, we have Z t Z t un (t) = x + ty + A (t − s)un (s)ds + (t − s)fn (s)ds (t ∈ R). 0

0

Since A Ris closed and un , fn ∈ BC(R, X), n→∞ R t for a fixed t ∈ R, Rletting t t we have 0 (t−s)u(s)ds ∈ D(A), and A 0 (t−s)un (s)ds → A 0 (t−s)u(s)ds. Therefore, Z t Z t u(t) = x + ty + A (t − s)u(s)ds + (t − s)f (s)ds (t ∈ R). 0

0

Thus, by definition, u is a mild solution of Eq. (4.15).

Remark 4.4. By a similar argument, we can easily show that the part of L on AAΛ (here Λ is a closed subspace of R) that is denoted by LΛ is a closed linear operator. Proposition 4.1. Let A be a closed linear operator. Then every weak solution of Eq. (4.15) is a mild solution. Proof. The proof is obvious in view of the above lemma and the remark that classical solutions are mild solutions. 4.5.2

Operators A

Let Λ be a closed subset of R. We first consider the operator AΛ of multiplication by A and the differential operator d/dt on the function space AAΛ (X). By definition the operator AΛ of multiplication by A is defined on D(AΛ ) := {g ∈ AAΛ (X) : g(t) ∈ D(A) ∀t ∈ R, Ag(·) ∈ AAΛ (X)}, and Ag := Ag(·) for all g ∈ D(AΛ ). In the following result we assume that Λ is a closed subset of the real 2 defined in AAΛ (X) with line and the second order differential operator DΛ 2 domain D(DΛ ) consisting of all functions u ∈ AAΛ (X) that are of class C 2 such that d2 u/dt2 ∈ AAΛ (X). Let us define −Λ2 := {ξ ∈ R|ξ = −λ2 , for some λ ∈ Λ}. Proposition 4.2. With the above notations the following assertions hold true 2 σ(DΛ ) = −Λ2 . (4.19)

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We associate with the equation

d2 u = µu + f (t), f ∈ AAΛ (X) dt2 the following first order equation n x01 = x2 x02 = µx1 + f (t), f ∈ AAΛ (X).

(4.20)

(4.21)

It is easily seen from the theory of ODE that the solvability of these equations in BC(R, X) are equivalent. As shown in Theorem 4.10, if f ∈ AAΛ (X) and iΛ ∩ σ(I(µ)) = ,

(4.22)

where I(µ) denotes the operator matrix associated with Eq.(4.21). A simple computation shows that σ(I(µ)) consists of all solutions to the equation t2 − µ = 0. Thus, 2 σ(DΛ ) ⊂ {µ ∈ C : µ = −λ2 for some λ ∈ Λ}.

2 Hence σ(DΛ ) ⊂ −Λ2 .

On the other hand, let µ ∈ Λ. Then g(·) := xeiµ· ∈ AAΛ (X). Obviously, 2 2 = −µ2 g and thus, −µ2 ∈ σ(DΛ ). Therefore, σ(DΛ ) ⊃ −Λ2 . That is, the proposition is proven. 2 DΛ g

Proposition 4.3. Let Λ be a closed subset of the real line and let Σ(ε) denote the set {z ∈ C|z 6= 0, | arg z| < π − ε}, where ε is a given small positive number. Then the resolvent R(λ, D 2 ) of the operator D 2 in AAΛ (X) satisfies M , ∀λ ∈ Σ(ε), (4.23) kR(λ, D2 )k ≤ |λ| where M is a positive constant depending only on ε. Proof.

Consider the equation x00 (t) − λx(t) = f (t),

t ∈ R,

(4.24)

for every given λ ∈ Σ(ε) and f ∈ AAΛ (X). As shown in the previous proposition, there exists a unique function xf ∈ AAΛ (X) that satisfies the above equation. For λ ∈ Σ(ε) there are exactly two distinct solutions λ1 , λ2 = −λ1 of the equation x2 − λ = 0. Without loss of generality, we assume that <eλ1 < 0. It is easy to check that the unique bounded solution to Eq. (4.24) is given by Z t Z ∞ 1 λ1 (t−ξ) −λ1 (t−ξ) xf (t) = e f (ξ)dξ + e f (ξ)dξ (4.25) 2λ1 −∞ t

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that is in AAΛ (X). By definition, xf = −R(λ, D2 )f . Obviously, since λ ∈ Σ(ε), we have kxf k ≤

1 2 M · ≤ , 2|λ1 | <eλ1 |λ|

where M is a positive constant depending on ε.

Proposition 4.4. Let A be a closed linear operator such that σ(A)∩−Λ 2 = . Then for every f ∈ BC(R, X) with sp(f ) ⊂ Λ, Eq. (4.15) has at most one mild solution u ∈ BC(R, X). Proof. Since the operator L is a closed linear operator, it suffices to show that if u ∈ BC(R, X) is a mild solution of Eq. (4.15) with f replaced by 0, then sp(u) = . This can be checked directly as follows. Assume that u is a solution of Eq. (4.17) with f replaced by 0. Taking Carleman-Laplace transforms of both sides of Eq. (4.17) we have y u ˆ(λ) x (<λ 6= 0). u ˆ(λ) = + 2 + A 2 λ λ λ Therefore, (λ2 − A)ˆ u(λ) = λx + y,

(<λ 6= 0).

(4.26)

If ξ ∈ R and ξ 6∈ σ(A), then (ξ 2 − A) is invertible and (λ2 − A)−1 is holomorphic in a small neighborhood of ξ. Therefore, u ˆ(λ) has a holomorphic extension to a neighborhood of ξ, so sp(u) ⊂ R ∩ σ(A). On the other hand, by the assumption that sp(u) ⊂ −Λ2 and −Λ2 ∩ σ(A) = , we have sp(u) = . And hence, u = 0. Lemma 4.5. Let A be the generator of an analytic semigroup. Then the 2 operator AΛ of multiplication by A and the differential operator DΛ on AAΛ (X) are commuting and satisfy condition P of Definition 1.11. Proof. Let us denote by T (t) the semigroup generated by A. As shown in the previous section 4.4, the operator A of multiplication by A generates an analytic semigroup T (t) of multiplication by T (t) on AAΛ (X). So, it is sufficient to prove that the semigroup T (t) commutes with D 2 , that is T (t)D(D2 ) ⊂ D(D2 ) and T (t)D2 x = D2 T (t)x,

∀x ∈ D(D2 ), t ≥ 0.

By the definition of the operator T (t) of multiplication by T (t), the above claim is obvious. The lemma now follows from the above propositions.

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So, by the spectral properties of sums of commuting operators, we have Corollary 4.4. If A generates an analytic semigroup with σ(A)∩−Λ2 = , then for every f ∈ AAΛ (X) there exists a unique u ∈ AAΛ (X) such that 2 − A u = f. DΛ Λ

2 Proof. Since AΛ and −DΛ commute and satisfy Condition P, the sum 2 2 − A ). From σ(A) ∩ −Λ2 = DΛ − AΛ is closable (denote its closure by DΛ Λ 2 − A ). and Theorem 1.11 in Appendix, it turns out that 0 ∈ ρ(DΛ Λ 2 −A ) Therefore, for every f ∈ AAΛ (X) there exists a unique u ∈ D(DΛ Λ such that 2 − A u = f. DΛ Λ

Now our remaining task is just to explain what the above closure means. More precisely, we will relate it with the notion of mild solutions to evolution equations. 2 − A ) and D 2 − A u = f , Lemma 4.6. Let u, f ∈ AA(X). If u ∈ D(DΛ Λ Λ Λ then u is a mild solution of Eq. (4.15).

Proof. The lemma follows from the fact that weak solutions are mild solutions. As an immediate consequence of the above argument we have: Theorem 4.10. Let A be the generator of an analytic semigroup and let Λ be a closed subset of R. Then it is necessary and sufficient for each f ∈ AAΛ (X) there exists a unique mild solution u ∈ AAΛ (X) to Eq. (4.15) that the condition σ(A) ∩ −Λ2 = holds. Proof. The sufficiency follows from the above argument. The necessity can be shown as follows: For every ξ ∈ Λ, obviously that the function h : R 3 t 7→ aeiξt is in AAΛ (X), where a ∈ X is any given element. By assumption, there is a unique g ∈ D(AΛ ) such that −ξg(t) − Ag(t) = h(t) for all t ∈ R. Following the argument in p. 252 of [Murakami, Naito and Minh (71)] one can easily show that g(t) is of the form beiξt . Hence, b is the unique solution of the equation −ξb − Ab = a. That is −ξ 6∈ σ(AΛ ), so −Λ ∩ σ(AΛ ) = . Corollary 4.5. Let A be the generator of an analytic semigroup such that σ(A) ∩ [−spu (f )]2 = . Then Eq. (4.15) has a unique almost automorphic mild solution w such that spu (w) ⊂ spu (f ).

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Proof.

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Set Λ = spu (f ). Then by the above argument we get the theorem.

Remark 4.5. We notice that all results stated above for almost automorphic solutions hold true for compact almost automorphic solutions if the assumption on the almost automorphy of f is replaced by the compact almost automorphy of f . Details of the proofs are left to the reader. 4.5.3

Nonlinear Equations

In this subsection we consider nonlinear equations of the form d2 u(t) = Au(t) + F (t, u(t)), t ∈ R, (4.27) dt2 where F : R×X → X is assumed to satisfies the following conditions: F (t, x) is almost automorphic in t for every fixed x ∈ X and F is continuous jointly in (t, x), Lipschtz in x uniformly in t, that is, there is a positive number ε independent of t, x such that kF (t, x) − F (t, y)k ≤ εkx − yk, ∀x, y ∈ X. (4.28) We will say that Condition H holds for F in AAΛ (X) if for a closed subset Λ ⊂ R the Nemystky operator F, defined by Fg(t) := F (t, g(t)), ∀g ∈ AAΛ (X), t ∈ R is an operator AAΛ (X) → AAΛ (X). Lemma 4.7. If F (t, x) is almost automorphic in t for every fixed x ∈ X, F is continuous jointly in (t, x) and satisfies (4.28), then the Nemystky operator F is a continuous operator acting on AA(X), that is, condition H holds for F on AAR (X). Proof.

For the proof see Theorem 2.2.6 in [N’Gu´er´ekata (79)].

Remark 4.6. In general, we are still puzzled with conditions on F and Λ such that condition H holds for F on AAΛ (X). In the linear case, if F (t, x) = L(t)x is T -periodic in t and Λ = {ξ ∈ R|eiξ ∈ G}, where G is a closed subset of the unit circle, then condition H holds for F and Λ (see e.g. [Batty, Hutter and R¨ abiger (14)]). In the general nonlinear case, this question is still open. Definition 4.4. A continuous X-valued function u is called a mild solution on R of (4.27) if Z t (t − s)u(s)ds ∈ D(A), ∀t ∈ R (4.29) 0

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and u(t) = x + ty + A

Z

t 0

(t − s)u(s)ds +

Z

t 0

(t − s)f (s, u(s))ds (t ∈ R) (4.30)

for some fixed x, y ∈ X. If u ∈ C 2 (R, X), u(t) ∈ D(A), ∀t ∈ R and Eq.(4.27) holds for all t ∈ R we say that u is a classical solution to Eq.(4.27). It is easily seen that if u ∈ C(R, X) is a classical solution of Eq. (4.27), then it is a mild solution of Eq. (4.27). The following is a main result of this subsection: Theorem 4.11. Let A be the generator of an analytic semigroup, and let condition H hold for F on AAΛ (X). Furthermore, we assume that σ(A) ∩ −Λ2 = and the Lipshitz coefficient ε in (4.28) is sufficiently small. Then, Eq. (4.27) has a unique almost automorphic mild solution u ∈ AAΛ (X). Proof. First we consider the operator LΛ defined in Definition 4.3 and the remark that follows. Under the assumptions, by Theorem 4.10, this operator is invertible on AAΛ (X). Next, we consider the operator LΛ − F. This operator may be seen as a nonlinear perturbation of LΛ . Since LΛ is a closed linear operator, if ε is sufficiently small, and if we consider D(LΛ ) ⊂ AAΛ (X) with the graph norm of LΛ , then by the Inverse Function Theorem, the operator LΛ − F is an invertible Lipschitz operator from D(LΛ ) onto AAΛ (X). Hence, Eq. (4.27) has a unique almost automorphic mild solution. 4.6

The Equations x’=f(t,x)

In this section we are going to study the existence of almost automorphic solutions of nonlinear differential equations and characterize some topological properties of the almost automorphic solutions sets . Thus we consider in a Banach space X the Cauchy problem CP x0 = f (t, x), where x0 ∈ X and f : R × X → X. Let us recall the following:

x(0) = x0 ,

t∈R

Definition 4.5. Let X, Y be metric spaces, f : X → Y be a continuous function and let y ∈ Y . f is said to be proper at the point y provided that there exists ε > 0 such that for any compact set K ⊂ B(y, ε) the set

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f −1 (K) is compact, where B(y, ε) is the open ball in Y of center y and radius ε. If f −1 (K) is compact for any compact K ⊂ Y , then f is called proper. Now we denote by α the Kuratowski measure of noncompactness and assume the following: 10 f is a continuous mapping such that kf (s, x)k ≤ φ(s)

+∞ Z φ(s)ds < +∞;

and

−∞

0

2 for each bounded subset M ⊂ X and for each interval [min(0, a), max(0, a)], there exists a continuous nondecreasing function hM,a : R+ → R+ such that the inequality u(t) ≤

max(0,t) Z

hM,a (u(s))ds,

for t ∈ [min(0, a), max(0, a)],

min(0,t)

has only a trivial solution u ≡ 0 and α(f (A × E)) ≤ hM,a (α(E)) for A ⊂ [min(0, a), max(0, a)] and E ⊂ M . Now we start with the following: Theorem 4.12. Under the above assumptions the set S of all solutions of the above CP, defined on R and considered as a subset of the space C = Cb (R, X) of all bounded continuous functions R → X with the topology of uniform convergence, is an Rδ . Proof.

Define the mapping Zt F (x)(t) = x0 + f (s, x(s))ds 0

for t ∈ R and x ∈ C.

Obviously F (C) ⊂ C. In view of the inequalities kF (x)(t1 )−F (x)(t2 )k ≤

Zt2

t1

kf (s, x(s))kds ≤

Zt2

t1

φ(s)ds

for t1 , t2 ∈ R, x ∈ C

and from the assumption about φ, it is clear that the family F (C) is equicontinuous. Now we verify that F is a continuous operator. Let (xn ) be a

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sequence of elements from C such that xn → x uniformly as n → ∞. Fix ε > 0. First consider the halfline R+ . Choose t0 > 0 such that +∞ R φ(s)ds < ε2 . In view of Krasnosielskii-Krein lemma there exists δ > 0 2 t0

such that

sup kf (s, u(s)) − f (s, x(s))k <

s∈[0,t0 ]

ε , 2t0

whenever u ∈ C and sup ku(s) − x(s)k < δ. Choose N ∈ N such that s∈[0,t0 ]

sup kxn (s) − x(s)k < δ for n ≥ N . For n ≥ N and t > t0 we have

s∈[0,t0 ]

kF (xn )(t) − F (x)(t)k = k Zt0 0

Zt 0

[f (s, xn (s)) − f (s, x(s))]dsk ≤

kf (s, xn (s)) − f (s, x(s))kds +

Zt

t0

kf (s, xn (s)) − f (s, x(s))kds <

ε ε + = ε, 2 2 so F (xn )|R+ → F (x)|R+ pointwise, as n → ∞. A similar reasoning establishes that F (xn )|R− → F (x)|R− pointwise, as n → ∞. Since the family F (C) is equicontinuous, we infer that F (xn ) → F (x) uniformly on R, as n → ∞, which proves the continuity of F . Define  x0 , if − n1 ≤ t ≤ n1 ,    1 t−   Rn  x0 + f (s, x(s))ds, if t > n1 , Fn (x)(t) = 0  1  t+  Rn    x0 + f (s, x(s))ds, if t < − n1 , 0

where x ∈ C and n ∈ N. Obviously Fn maps C into itself. Essentially the same reasoning as in the case of F proves that mappings Fn are continuous. Put Tn = I − Fn for n ∈ N, where I denotes the identity map on C. The mappings Tn are continuous. We establish that they are homeomorphisms. Indeed, fix n ∈ N. It is easy to see that for any x1 , x2 ∈ C: Tn (x1 ) = Tn (x2 ) =⇒ x1 = x2 .

Now, it is enough to prove the continuity of Tn−1 . Suppose (xi ) is a sequence of elements from C such that Tn (xi ) → Tn (x) uniformly as i → ∞. Since

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Fn (xi )(t) = x0 = Fn (x)(t) for |t| ≤ n1 , xi → x uniformly on {t : |t| ≤ n1 }, as i → ∞. Further, applying the Krasnoselskii-Krein lemma, we infer that 1 t− Z n

0

1 t− Z n

f (s, xi (s))ds →

f (s, x(s))ds

0

for

2 1
1 n

uniformly, so xi → x on {t : < t ≤ n2 } uniformly, as i → ∞. The similar reasoning establishes that xi → x uniformly on [0, t0 + n1 ], as i → ∞. For t > t0 + n1 we have 1 t− n

Z

kFn (xi )(t) − Fn (x)(t)k = k Zt0 0

[f (s, xi (s)) − f (s, x(s))]dsk ≤

0

kf (s, xi (s)) − f (s, x(s))kds +

1 t− Z n

t0

kf (s, xi (s)) − f (s, x(s))kds

The above inequalities and similar arguments as in the case of the continuity of F prove that Fn (xi ) → Fn (x) uniformly on [t0 + n1 , +∞), so xi → x uniformly on this interval, as i → ∞. In a similar manner we argue in the case of the halfline R− . Hence xi → x uniformly on R, as i → ∞. This proves the continuity of Tn−1 . Now we shall show that lim Tn = T uniformly. We have the following n→∞ inequalities kFn (x)(t) − F (x)(t)k = k for x ∈ C

and |t| ≤

Zt 0

1

f (s, x(s))dsk ≤

Zn

1 , n

kFn (x)(t) − F (x)(t)k ≤

Zt

kf (s, x(s))kds ≤

and t >

In particular, for t > t0 +

1 n

Zt

1 t− n

1 t− n

for x ∈ C

φ(s)ds

0

1 . n

we have

kFn (x)(t) − F (x)(t)k ≤

Zt

1 t− n

φ(s)ds <

ε . 4

φ(s)ds

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Thus Fn (x) → F (x) uniformly in x on R+ . Arguing in a similar way we infer that Fn (x) → F (x) uniformly in x on R. To complete the proof it is enough to show that T is proper at 0 ∈ C. Let Z be any compact set in C and put U = T −1 (Z). Consider the sequence (un ), where un ∈ U for n ∈ N. Set V = {un : n ∈ N}. Since V (t) ⊂ T (V )(t) + F (V )(t) ⊂ Z(t) + F (V )(t) and Z(t) is compact, in view of the properties of the index α we obtain α(V )(t) ≤ α(Z(t)) + α(F (V )(t)) = α(F (V )(t))

for t ∈ R.

Thus the similar reasoning as in the proof of Theorem 3 [Bugajewska (16)] proves that V (t) is compact for every t ∈ R. In view of Ascoli’s theorem we infer that V is relatively compact and thus (un ) has a limit point. Therefore U is a compact, so T is proper. In view of Theorem 6.7, T −1 (0) is an Rδ set. Since T −1 (0) coincides with the set of all fixed points of F , i.e. it coincides with S, the proof is complete.

Now assume 30 (s, x) → f (s, x) is a function from R×X to X, which is almost automorphic in s for each x and satisfies a Lipschitz condition in x uniformly in s ∈ R; 40 for each x ∈ AA(X) the range RF (x) = {F (x)(s) : s ∈ R}, where F is the operator defined in the proof of the above theorem, is relatively compact. We now give the following important result: Theorem 4.13. Suppose 20 , 30 , 40 are satisfied and that the function f satisfies the inequality as in 10 . Then CP has a solution in AA(X). Moreover, the set S of all such solutions is compact. Proof. In view of 30 and [N’Gu´er´ekata (79)] Theorem 2.2.6, the function s → f (s, x(s)), s ∈ R is almost automorphic for any x ∈ AA(X). By 40 and [N’Gu´er´ekata (79)], Theorem 2.4.4, F (x) ∈ AA(X) for any x ∈ AA(X). Hence F maps AA(X) into itself. Let D = conv(F (AA(X))). It is clear that F maps D into itself. Essentially the same reasoning as in the proof of Theorem 3 proves that the family F (AA(X)) is equicontinuous and that F is a continuous mapping. Now, let V be a subset of D such that V ⊂

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conv(F (V ) ∪ {0}), where 0 ∈ AA(X). In view of the properties of the index α we obtain α(V (t)) ≤ α(convF (V )(t) ∪ {0}) = α(F (V )(t) ∪ {0}) =

max(α(F (V )(t)), α({0})) = α(F (V )(t))

for t ∈ R,

and therefore the same reasoning as in the proof of Theorem 3 [Bugajewska (16)] proves that V (t) is relatively compact for t ∈ R. In view of Ascoli’s theorem we infer that V is relatively compact. The operator F satisfies all assumptions of Theorem 1 and, therefore, there exists z ∈ D such that z = F (z). Finally, since S = F (S), the same reasoning as in Theorem 3 [Bugajewska (16)] proves that S is relatively compact. Since S is closed, it is compact, which completes the proof. Remark 4.7. In the case of uniformly convex Banach spaces the assumption 40 in Theorem 4 can be weaken. Namely, in view of [N’Gu´er´ekata (79)], Theorem 2.4.6 it is enough to assume then that the range RF (x) is bounded for each x ∈ AA(X). Remark 4.8. By the assumption 30 and [N’Gu´er´ekata (79)] Theorem 2.2.6, the function s → f (s, x(s)), s ∈ R is almost automorphic for each x ∈ AA(X). Thus by [N’Gu´er´ekata (79)] Theorem 2.2.6 the range {f (s, x(s)) : s ∈ R} is relatively compact in X for each x ∈ AA(X). Further, since F (x) is almost automorphic if and only if the range RF (x) is relatively compact in X ([N’Gu´er´ekata (79)], Theorem 2.4.4.), the assumption 40 is natural in the considered situation. 4.7

Comments and Further Reading Guide

Recently, numerous contributions were made to the study of abstract evolution equations with almost automorphic solutions. These include work by D. Bugajewski, T. Diagana, S. G. Gal, S. C. Gal, J. A. Goldstein, K. Ezzinbi, Y. Hino, J. Liu, S. Murakami, G. M. N’Gu´er´ekata, Nguyen Van Minh, A. S. Rao, S. Zaidman, etc. Note that Theorem 4.2 is usually referred as Bohr-Neugebauer type theorem. The proof presented here is a slight modification of Theroem 4.2.2 [N’Gu´er´ekata (79)]. There exist several generalizations of the BohrNeuhegauer Theorem. We send the reader for instance to: [N’Gu´er´ekata (81)], [N’Gu´er´ekata (83)], [N’Gu´er´ekata (88)].

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Existence of almost automorphic solutions to differential equations is a fundamental problem still under investigation. Various methods were used including fixed point theorems for linear equations x0 = Ax(t) + f (t) ([N’Gu´er´ekata (87)]), as well as nonlinear and inte-grodifferential equations ([Bugajewski and N’Gurkata (17)]) and semilinear equation x0 = Ax(t) + f (t, x) ([N’Gu´er´ekata (79)], [N’Gu´er´ekata (88)]) when A generates an asymptotically stable semigroup of linear operators; the method of invariant subspaces, introduced in 2003 by T. Diagana and G. M. N’Gu´er´ekata for perturbed equations of the form x0 = (A + B)x + f (t) where both operators A and B are unbounded; the method of sums of commuting operators (Theorems 4.9, 4.11). In this latter case, the challenge is to overcome the difficulty due to the fact that almost automorphic functions are not necessarily uniformly continuous. This implies the non-strong continuity of translation semigroup in the function space AA(X). This approach, as well as the method of invariant subspaces are also used successfully in the case of higher order differential equations in various recent papers by T. Diagana, J. Liu, G. M. N’Gu´er´ekata, and Nguyen van Minh. Readers interested in fuzzy settings of the theory of almost automorphy may consult [N’Gu´er´ekata (88)] and other papers by S. G. Gal and G. M. N’Gu´er´ekata.

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Chapter 5

Nonlinear equations

5.1

Periodic Solutions of Nonlinear equations

5.1.1

Nonlinear Equations Without Delay

In this section, we study the existence of periodic solutions of the following nonlinear differential equation without delay in a general Banach space X = (X, k · k), u0 (t) + A(t)u(t) = f (t, u(t)), t > 0, u(0) = u0 .

(5.1)

To this end, we make the following assumptions. Assumption 5.1. For a constant T > 0, A(t + T ) = A(t), t ≥ 0. The function f is continuous in all its variables, T -periodic in the first variable t and Lipschitzian in the other variables uniformly in t. Remark 5.1. We used “variable(s)” in the assumption 5.1 because then it can be used for other cases where the function f is of more than two variables. Assumption 5.2. For t ∈ [0, T ], (H1). The domain D(A(t)) = D is independent of t and is dense in X. (H2). For t ≥ 0, the resolvent R(λ, A(t)) = (λI − A(t))−1 exists for all λ with Reλ ≤ 0 and is compact, and there is a constant M independent of λ and t such that kR(λ, A(t))k ≤ M (|λ| + 1)−1 , Reλ ≤ 0. (H3). There exist constants L and 0 < a ≤ 1 such that k(A(t) − A(s))A(r)−1 k ≤ L|t − s|a , s, t, r ∈ [0, T ]. 153

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Under these assumptions, the results in, e.g., Pazy [Pazy (90)], imply the existence of a unique evolution system U (t, s) (0 ≤ s ≤ t ≤ T ) for Eq. (5.1). The materials presented here are based on [Liu (60)]. Theorem 5.1. Let the assumptions 5.1 and 5.2 be satisfied and let u0 ∈ X. Then there exists a constant α > 0 such that Eq. (5.1) has a unique mild solution (also called solution here) u : [0, α] → X satisfying Z t u(t) = U (t, 0)u0 + U (t, h)f (h, u(h))dh, t ∈ [0, α]. (5.2) 0

Proof. We only need to set up the framework for the use of the contraction mapping principle. With u0 ∈ X being fixed and with α > 0 yet to be determined, we define a map (operator) Q on C([0, α], X) such that Z t U (t, h)f (h, u(h))dh, for t ∈ [0, α]. (5.3) (Qu)(t) = U (t, 0)u0 + 0

Using the property of the evolution system U , we have Q : C([0, α], X) → C([0, α], X). Next, for u, v ∈ C([0, α], X), one has Z t h i U (t, h) f (h, u(h)) − f (h, v(h)) dh. (5.4) (Qu)(t) − (Qv)(t) = 0

Now, f is Lipschitzian in the second variable and U (t, h) is a bounded operator, it is then clear that we can obtain the result by using the contraction mapping principle. Details are left in an exercise. Note that we are concerned with periodic solutions here, so we assume that solutions exist on [0, ∞). We will write u = u(·, u0 ) for the unique solution with the initial value u0 . Next, we give some basic results concerning the search of periodic solutions. Lemma 5.1. Let the assumptions 5.1 and 5.2 be satisfied. (a). If u(t) is a solution of Eq. (5.1), then so is u(t + T ), t ≥ 0. (b). Let u(t, u0 ) be a solution of Eq. (5.1) with u(0, u0 ) = u0 . Then u(t, u0 ) is T -periodic if and only if u(T, u0) = u0 . Proof. (a): Let y(t) = u(t + T ). Now, for t ≥ 0, we can use the known formulas ([Pazy (90)]) U (t, s) = U (t, r)U (r, s), 0 ≤ s ≤ r ≤ t ≤ T , and U (t + T, s + T ) = U (t, s) (since the operator A(t) is T -periodic in t) to obtain:

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y(t) = u(t + T ) = U (t + T, 0)u0 +

= U (t + T, T )U (T, 0)u0 +

+

Z

Z

Z

155

t+T

U (t + T, h)f (h, u(h))dh 0

T

U (t + T, h)f (h, u(h))dh 0

t+T

U (t + T, h)f (h, u(h))dh

T

= U (t, 0)U (T, 0)u0 +

+

Z

Z

T

U (t + T, T )U (T, h)f (h, u(h))dh 0

t

U (t + T, T + s)f (T + s, u(T + s))ds

0

= U (t, 0)U (T, 0)u0 +

+

Z

Z

T

U (t, 0)U (T, h)f (h, u(h))dh 0

t

U (t, s)f (s, y(s))ds

0

h

= U (t, 0) U (T, 0)u0 +

+

Z

Z

T

U (T, h)f (h, u(h))dh 0

i

t

U (t, s)f (s, y(s))ds

0

= U (t, 0)u(T ) +

= U (t, 0)y(0) +

Z

Z

t

U (t, s)f (s, y(s))ds 0 t

U (t, s)f (s, y(s))ds.

(5.5)

0

The equality (5.5) implies that y(t) = u(t + T ) is also a solution of Eq. (5.1). (b): If u(t, u0 ) is a T -periodic solution of Eq. (5.1), then u(T, u0 ) = u(0, u0 ) = u0 . On the other hand, if u(T, u0 ) = u0 , then, from (a), y(t) = u(t + T, u0 ) is also a solution of Eq. (5.1) with y(0) = u(T, u0 ) = u0 . By

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uniqueness, y(t) = u(t, u0 ), or u(t + T, u0 ) = u(t, u0 ), t ≥ 0, thus u(t, u0 ) is T -periodic. This completes the proof. Accordingly, we can define a map P : X → X (called a Poincar´e map or a Poincar´e operator) such that for each u0 ∈ X and for the corresponding unique solution u(t) = u(t, u0 ) with u(0, u0 ) = u0 , define Z T U (T, r)f (r, u(r, u0 ))dr. (5.6) P (u0 ) = u(T ) = U (T, 0)u0 + 0

Now, Lemma 5.1 (b) indicates that Eq. (5.1) has a T -periodic solution if and only if there exists an u0 ∈ X such that P (u0 ) = u0 . This is the same as saying that the Poincar´e map P has a fixed point. We formulate this as follows. Lemma 5.2. Eq. (5.1) has a T -periodic solution if and only if the Poincar´e map P : X → X defined in (5.6) has a fixed point. We will see that Lemma 5.2 provides a very useful approach for deriving periodic solutions, as it reduces the search of periodic solutions to that of fixed points of the Poincar´e map P , for which some well-known fixed point theorems from Functional Analysis can be applied. By Massera’s theorems for nonlinear differential equations in
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integer m, one has P j (E1 ) ⊂ E2 , 1 ≤ j ≤ m − 1,

(5.7)

P j (E1 ) ⊂ E0 , m ≤ j ≤ 2m − 1,

(5.8)

then P has a fixed point in E2 .

Definition 5.1. [Burton (18)] The solutions of Eq. (5.1) are said to be bounded if for each B1 > 0 there is a B2 = B2 (B1 ) > 0 such that {ku0 k ≤ B1 , t ≥ 0} implies ku(t, u0 )k < B2 . Definition 5.2. [Burton (18)] The solutions of Eq. (5.1) are said to be ultimate bounded if there is a bound B > 0 such that for each B3 > 0, there is a K = K(B, B3 ) > 0 such that {ku0 k ≤ B3 , t ≥ K} implies ku(t, u0 )k < B. Definition 5.3. An operator S : X → X is said to be a compact operator on X if S is continuous and maps a bounded set into a precompact set. For the relationship between boundedness and ultimate boundedness, we have the following result. Theorem 5.3. Assume that f (t, u) is continuous and is Lipschitzian in u. If the solutions of Eq. (5.1) are ultimately bounded, then they are also bounded. Proof. Let B > 0 be the bound in the definition of ultimate boundedness, then for any B1 > 0, there is a K = K(B, B1 ) > 0 such that {ku0 k ≤ B1 , t ≥ K} imply ku(t, u0)k ≤ B. Next, for t in the interval [0, K], we have Z t kU (t, s)f (s, u(s, u0 ))kds ku(t, u0 )k ≤ kU (t, 0)u0 k+ 0

≤ kU (t, 0)u0 k+ +

Z

Z

0

+

kU (t, s)[f (s, u(s, u0 )) − f (s, 0)]kds

t

kU (t, s)f (s, 0)kds

0

≤ kU (t, 0)u0 k + c Z

t

Z

t 0

|U (t, s)|ku(s, u0 )kds

t 0

kU (t, s)f (s, 0)kds,

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where c is a Lipschitz constant. For 0 ≤ s ≤ t ≤ K, there are positive constants G1 and G2 such that Z t |U (t, s)| ≤ G1 , |U (t, 0)|B1 + kU (t, s)f (s, 0)kds ≤ G2 . 0

Then for t ∈ [0, K] and ku0 k ≤ B1 , Z t ku(t, u0 )k ≤ G2 + c G1 ku(s, u0)kds. 0

Hence, Gronwall’s inequality implies, for t ∈ [0, K] and ku0 k ≤ B1 , ku(t, u0 )k ≤ G2 ecG1 t ≤ G2 ecG1 K ,

(5.9)

which implies that for ku0 k ≤ B1 , the solutions u(t, u0 ) are bounded on [0, K]. That is, there is a B3 = B3 (B1 , K) = B3 (B1 , B) > 0 such that ku(t, u0 )k ≤ B3 for t ∈ [0, K] and ku0 k ≤ B1 . Now, we see that {ku0 k ≤ B1 , t ≥ 0} imply ku(t, u0 )k ≤ B2 = max{B3 , B}, which verifies the boundedness of the solutions. This completes the proof. To derive periodic solutions, the following lemma from Amann [Amann (2)] will be used to show that the Poincar´e map P defined in (5.6) is a compact operator. Recall that in the usual way (see, e.g., Amann [Amann (2)], Pazy [Pazy (90)]) we define fractional power operator Aα and Banach space Xα for 0 ≤ α ≤ 1, where A = A(0) and Xα = (D(Aα ), k · kα ) with kxkα ≡ kAα xk . We also write the norm in L(Xα , Xβ ) (space of bounded linear operators from Xα to Xβ ) as k · kα,β . Lemma 5.3. [Amann (2)] (i) Suppose that 0 ≤ α ≤ β < 1. Then for β − α < γ < 1, there is a constant C(α, β, γ) such that kU (t, h)kα,β ≤ C(α, β, γ)(t − h)−γ , 0 ≤ h < t ≤ T. (ii) For 0 ≤ γ < 1, there is a constant C(γ), such that for g ∈ C([0, L], X) (L > 0 is a constant), one has for 0 ≤ s, t ≤ L, Z t Z s k U (t, h)g(h)dh − U (s, h)g(h)dhk ≤ C(γ)|t − s|γ max kg(h)k. 0

0≤h≤L

0

(iii) Let 0 ≤ α < β ≤ 1. Then

K(x, g)(t) ≡ U (t, 0)x +

Z

t 0

U (t, h)g(h)dh, 0 ≤ t ≤ T,

defines a continuous linear operator from Xβ × C([0, T ], X) into C γ ([0, T ], Xα ) for every γ ∈ [0, β − α).

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Now we prove that the Poincar´e map P : X → X defined in (5.6) is compact. Theorem 5.4. Let assumptions 5.1 and 5.2 be satisfied and assume that solutions of Eq. (5.1) are bounded. Then the Poincar´e map P : X → X defined in (5.6) is a compact operator. Proof. The continuity of P is left as an exercise. Next, fix η ∈ (0, 1). Then from Lemma 5.3 (i), U (T, 0) : X → Xη is bounded. Next, let E be bounded in X. Since solutions of Eq. (5.1) are bounded, there exists M1 = M1 (E, f ) > 0 such that kf (t, u(t, u0))k ≤ M1 , t ∈ [0, T ], u0 ∈ E. Now for u0 ∈ E, we let g(t) = f (t, u(t, u0 )). Then Z T U (T, r)f (r, u(r, u0 ))dr ≡ K(0, g)(T ) ∈ Xη 0

according to Lemma 5.3 (ii). Also note that by Lemma 5.3 (i), there are constants γ ∈ (0, 1) and M2 > 0 such that kU (T, s)k0,η ≤ M2 (T − s)−γ , 0 ≤ s < T. Thus k

Z

0

T

U (T, r)f (r, u(r, u0 ))drkη ≤ M1 M2 T 1−γ /(1 − γ), u0 ∈ E.

Therefore P : X → Xη is bounded. Next, the embedding Xη → X is compact (see, e.g., [Hale (40)]), thus P : X → X is a compact operator. This completes the proof. Now, we are ready to state and prove the existence of periodic solutions for Eq. (5.1) without delay. Theorem 5.5. Let the assumptions 5.1 and 5.2 be satisfied. If the solutions of Eq. (5.1) are ultimate bounded, then Eq. (5.1) has a T -periodic solution. Proof. From Theorem 5.3, the solutions of Eq. (5.1) are also bounded. Let B be the bound in the definition of ultimate boundedness, then by boundedness, there is a B1 > 0 such that ku0 k ≤ B implies ku(t, u0 )k < B1 for t ≥ 0. Furthermore, there is a B2 > B1 such that ku0 k ≤ B1 implies ku(t, u0 )k < B2 for t ≥ 0. Now, using ultimate boundedness, there is a positive integer m such that ku0 k ≤ B1 implies ku(t, u0 )k < B for t ≥ (m − 1)T .

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We know from Lemma 5.1 that u(T + t, u0 ) is also a solution, then the uniqueness implies that u(T + t, u0 ) = u(t, u(T, u0 )), t ≥ 0. Letting t = T , this means P 2 (u0 ) = u(2T, u0). By an induction, it can be shown (see an exercise) that P k (u0 ) = u(kT, u0 ), k ≥ 1.

(5.10)

Thus, we have the following kP j−1 (u0 )k = ku((j − 1)T, u0 )k < B2 , j = 1, 2, ..., m − 1 and ku0 k ≤ B1 , kP j−1 (u0 )k = ku((j − 1)T, u0 )k < B, j ≥ m and ku0 k ≤ B1 .

Now let

  H = {u0 ∈ X : ku0 k < B2 } , E2 = cov.(P (H)),     K = {u0 ∈ X : ku0 k < B1 } , E1 = K ∩ E2 ,      G = {u ∈ X : ku k < B} , E = cov.(P (G)), 0 0 0

(5.11)

where cov.(F ) is the convex hull of the set F defined by cov.(F ) = Pn Pn { i=1 λi fi : n ≥ 1, fi ∈ F, λi ≥ 0, i=1 λi = 1}. From Theorem 5.4, the operator P is compact. It is also known that a convex hull of a precompact set is also precompact, then we see that E0 , E1 and E2 are convex subsets of X with E0 and E2 compact subsets and E1 open relative to E2 . Next, from (5.11), we have P j (E1 ) ⊂ P j (K) = P P j−1 (K) ⊂ P (H) ⊂ E2 , 1 ≤ j ≤ m − 1, P j (E1 ) ⊂ P j (K) = P P j−1 (K) ⊂ P (G) ⊂ E0 , m ≤ j ≤ 2m − 1.

Consequently, we know from Horn’s fixed point theorem that the operator P has a fixed point u0 ∈ X. This completes the proof by using Lemma 5.2. Next, we provide a result which asserts that the existence of a proper Liapunov function implies boundedness and ultimate boundedness of solutions. Theorem 5.6. Assume that there exist functions (“wedges”) Wi , i = 1, 2, 3, with Wi : [0, ∞) → [0, ∞), Wi (0) = 0, Wi strictly increasing, and W1 (t) → ∞, t → ∞. Further, assume that there exists a (Liapunov) function V : X → < (reals) such that for some constant M > 0, when u is a solution of Eq. (5.1) with ku(t)k ≥ M , then (a). W1 (ku(t)k) ≤ V (u(t)) ≤ W2 (ku(t)k), and

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(b).

d dt V

161

(u(t)) ≤ −W3 (ku(t)k).

Then solutions of Eq. (5.1) are bounded and ultimate bounded. Proof. Let u(t) = u(t, u0 ) and let B1 > 0 be given with B1 ≥ M . Find B2 ≥ B1 with W1 (B2 ) = W2 (B1 ). If for some interval [t1 , t2 ] with t1 ≥ 0, ku(t1 )k = B1 , and ku(t)k ≥ B1 on [t1 , t2 ], then for t ∈ [t1 , t2 ], W1 (ku(t)k) ≤ V (u(t)) ≤ V (u(t1 )) ≤ W2 (ku(t1 )k) = W2 (B1 ) = W1 (B2 ),

(5.12)

and hence ku(t)k ≤ B2 for t ∈ [t1 , t2 ]. This implies that if ku0 k ≤ B1 , then ku(t)k ≤ B2 , t ≥ 0, which gives the boundedness. Next, find B ≥ M + 1 with W1 (B) = W2 (M + 1). Then, similar to (5.12), one shows, for t1 ≥ 0, n o ku(t1 )k = M + 1, ku(t)k ≥ M + 1, t ∈ [t1 , t2 ] n o ⇒ ku(t)k ≤ B, t ∈ [t1 , t2 ] .

(5.13)

Let B3 > 0 be given. We need to prove that there is a K > 0 such that if ku0 k ≤ B3 and t ≥ K, then ku(t)k ≤ B. According to (5.13), we only need to show that there is a K > 0 such that when ku0 k ≤ B3 , there is a t0 ∈ [0, K] with ku(t0 )k ≤ M + 1. (Because then if there is a t2 ≥ K with ku(t2 )k > B ≥ M + 1, we can find such a t1 < t2 that ku(t1 )k = M +1 and that (5.13) can be used to get ku(t2 )k ≤ B, a controdiction.) Now, if ku(t)k > M + 1 for t ≥ 0, then 0 < W1 (M + 1) ≤ V (u(t)) ≤ V (u0 ) − W3 (M + 1)t ≤ W2 (B3 ) − W3 (M + 1)t. This fails when t ≥ W2 (B3 )/W3 (M + 1). Thus we can choose K ≡ W2 (B3 )/W3 (M + 1) to finish the proof. Exercise 20. (1) Complete the proof of Theorem 5.1. (2) In the proof of Theorem 5.4, show that the Poincar´e map P : X → X defined in (5.6) is continuous. (3) In the proof of Theorem 5.5, use an induction to show that P k (u0 ) = u(kT, u0), k ≥ 1. (4) Prove that a convex hull of a precompact set is also precompact.

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5.1.2

Nonlinear Equations With Finite Delay

Consider the following nonlinear differential equation with finite delay, u0 (t) + A(t)u(t) = f (t, u(t), ut ), t > 0, u(s) = φ(s), s ∈ [−r, 0],(5.14)

in a general Banach space X, where r > 0 is a constant, and ut (s) = u(t + s), s ∈ [−r, 0]. The materials presented here are based on [Liu (61)]. We still assume that the assumptions 5.1 and 5.2 are satisfied, so similar to the previous section, we have Theorem 5.7. Let the assumptions 5.1 and 5.2 be satisfied and let φ ∈ C([−r, 0], X) (the Banach space with the sup-norm). Then there exists a constant α > 0 such that Eq. (5.14) has a unique (mild) solution u : [−r, α] → X satisfying u0 = φ (i.e., u(s) = φ(s), s ∈ [−r, 0]), and Z t u(t) = U (t, 0)φ(0) + U (t, h)f (h, u(h), uh )dh, t ∈ [0, α]. (5.15) 0

To derive periodic solutions, we assume that solutions exist on [0, ∞). We will write u = u(·, φ) for the unique solution with the initial function φ. Note that for equations without delay, the Poincar´e operator maps a single element of the Banach space X to a single element of X. Now, for equations with finite delay, the difference is that the Poincar´e operator will map an element φ of C([−r, 0], X) (thus a function) to an element of C([−r, 0], X), given by P φ = uT (·, φ), φ ∈ C([−r, 0], X),

(5.16)

or, for s ∈ [−r, 0], (P φ)(s) = uT (s, φ) = u(T + s, φ)

=

 Z T +s    U (T + s, 0)φ(0) + U (T + s, h)f (h, u(h), uh )dh,    0

T + s > 0,

     

φ(T + s), T + s ≤ 0.

Similar to lemmas 5.1 and 5.2, we have Lemma 5.4. Let the assumptions 5.1 and 5.2 be satisfied. (a). If u(t) is a solution of Eq. (5.14), then so is u(t + T ), t ≥ 0.

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(b). Equation (5.14) has a T -periodic solution if and only if the Poincar´e map P : C([−r, 0], X) → C([−r, 0], X) defined in (5.16) has a fixed point. Definitions concerning boundedness and ultimate boundedness given in the previous section can be revised as follows to suit Eq. (5.14). Definition 5.4. Denote C = C([−r, 0], X). The solutions of Eq. (5.14) are said to be bounded if for each B1 > 0 there is a B2 = B2 (B1 ) > 0 such that {|φ|C ≤ B1 , t ≥ 0} implies ku(t, φ)k < B2 . (Here | · |C means the sup-norm.) Definition 5.5. The solutions of Eq. (5.14) are said to be ultimate bounded if there is a bound B > 0 such that for each B3 > 0, there is a K = K(B, B3 ) > 0 such that {|φ|C ≤ B3 , t ≥ K} implies ku(t, φ)k < B. For the relationship between boundedness and ultimate boundedness, we also have the following result. Theorem 5.8. Assume that f (t, u, v) in Eq. (5.14) is continuous and is Lipschitzian in u and in v. If the solutions of Eq. (5.14) are ultimately bounded, then they are also bounded. From the study for equations without delay, we see that for Eq. (5.14) with finite delay, the most important step will be the proof of the compactness of the Poincar´e operator defined in (5.16). This is getting a little hard since now we need to deal with functions rather than single elements. The following Ascoli-Arzela theorem for general Banach spaces is needed here to treat a set of functions. Theorem 5.9. (Ascoli-Arzela) Let E ⊂ C([−r, 0], X) be bounded. Then E is precompact if and only if functions in E are equicontinuous and for each t ∈ [−r, 0], the set {f (t) : f ∈ E} is precompact in X. Accordingly, for a bounded set E ⊂ C([−r, 0], X), we need to prove the precompactness of [P (E)](s) = {(P φ)(s) : φ ∈ E}

for every s ∈ [−r, 0]. Now, note from geometry that the operator P defined in (5.16) maps an initial function φ on [−r, 0] to the function P φ on [T − r, T ] along the unique solution u(·, φ). If T − r ≤ 0, then the restrictions of P φ on [T − r, 0] for φ ∈ E are parts of the initial functions φ ∈ E, and they may be

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arbitrary, or “bad”, i.e., noncompact. Therefore, to avoid this, we require T − r > 0. Now, for φ ∈ C([−r, 0], X), P φ = uT (·, φ) is a function defined on [T − r, T ] ⊂ (0, ∞), thus the possibly “bad” history on [−r, 0] is cut off. Then the operator P can smooth things out so that the same idea for equations without delay and Horn’s fixed point theorem can be applied to derive periodic solutions. Theorem 5.10. Let the assumptions 5.1 and 5.2 be satisfied and let T > r. If the solutions of Eq. (5.14) are bounded, then P : C([−r, 0], X) → C([−r, 0], X) defined in (5.16) is a compact operator. Proof. The continuity of P is left as an exercise. Let H ⊂ C([−r, 0], X) be bounded. Since the solutions of Eq. (5.14) are bounded, it follows that E = P (H) ⊂ C([−r, 0], X) is bounded. In the following, we will use the Ascoli-Arzela theorem to show that E is precompact. Note that T − r > 0, so for s ∈ [−r, 0], a function in E can be expressed as (P φ)(s) = uT (s, φ) = u(T + s, φ) Z T +s = U (T + s, 0)φ(0) + U (T + s, h)f (h, u(h), uh )dh, φ ∈ H. 0

Also, as T − r > 0, there is a k > 0 such that T + s > k for s ∈ [−r, 0]. From the properties for the evolution system U (t, s), one has, for s ∈ [−r, 0], U (T + s, 0)φ(0) = U (T + s, k)U (k, 0)φ(0), φ ∈ H.

(5.17)

Fix η ∈ (0, 1). Then from Lemma 5.3 (i), U (k, 0) : X → Xη is bounded. Next the embedding Xη → X is compact, thus {U (k, 0)φ(0) : φ ∈ H} is precompact in X since {φ(0) : φ ∈ H} is bounded in X. Therefore, the closure of {U (k, 0)φ(0) : φ ∈ H} is compact in X. Now, one can verify that as functions on · ∈ [−r, 0], h i {U (T + ·, 0)φ(0) : φ ∈ H} = {U (T + ·, k) U (k, 0)φ(0) : φ ∈ H} (5.18)

is equicontinuous. Next, from Lemma 5.3 (ii), for 0 ≤ γ < 1, there is a constant C(γ), such that for s1 , s2 ∈ [−r, 0], Z T+s2 Z T +s1 k U (T + s2 , h)f (h, u(h), uh )dh − U (T + s1 , h)f (h, u(h), uh )dhk 0

0

≤ C(γ)|s1 − s2 |γ max kf (h, u(h), uh )k. 0≤h≤T

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Since solutions of Eq. (5.14) are bounded and f maps a bounded set into a bounded set, there exists M1 = M1 (H) > 0 such that kf (t, u(t, φ), ut (φ))k ≤ M1 , t ∈ [0, T ], φ ∈ H.

Thus, as functions on · ∈ [−r, 0], Z T +· { U (T + ·, h)f (h, u(h), uh )dh : φ ∈ H}

(5.19)

0

is also equicontinuous. Therefore, functions in E are equicontinuous. Next, to check the precompactness of the functions at every point of [−r, 0], we fix s0 ∈ [−r, 0]. From the above arguments, we also know that {U (T + s0 , 0)φ(0) : φ ∈ H}

(5.20)

is precompact in X. Now for φ ∈ H, we let g(t) = f (t, u(t, φ), ut (φ)). Then Z T +s0 U (T + s0 , h)f (h, u(h, φ), uh (φ))dh ≡ K(0, g)(T + s0 ) ∈ Xη 0

according to Lemma 5.3 (iii). Also note that by Lemma 5.3 (i), there are constants γ ∈ (0, 1) and M2 > 0 such that kU (T + s0 , h)k0,η ≤ M2 (T + s0 − h)−γ , 0 ≤ h < T + s0 .

Thus Z T +s0 k U (T +s0 , h)f (h, u(h, φ), uh (φ))dhkη ≤ M1 M2 T 1−γ /(1−γ), φ ∈ H. 0

Therefore

{

Z

T +s0 0

U (T + s0 , h)f (h, u(h, φ), uh (φ))dh : φ ∈ H}

(5.21)

is bounded in Xη . Then use the fact that the embedding Xη → X is compact again, we see that the set defined by (5.21) is precompact in X. Now the Ascoli-Arzela theorem implies that the map P is a compact operator. Similar to the case for equations without delay, we have Theorem 5.11. Let the assumptions 5.1 and 5.2 be satisfied and let T > r. If the solutions of Eq. (5.14) are ultimate bounded, then Eq. (5.14) has a T -periodic solution. Theorem 5.12. Assume that there exist functions (“wedges”) Wi , i = 1, 2, 3, with Wi : [0, ∞) → [0, ∞), Wi (0) = 0, Wi strictly increasing, and W1 (t) → ∞, t → ∞. Further, assume that there exists a (Liapunov) function V : X → < (reals) such that for some constant M > 0, when u is a solution of Eq. (5.14) with ku(t)k ≥ M , then

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(a). W1 (ku(t)k) ≤ V (u(t)) ≤ W2 (ku(t)k), and d (b). dt V (u(t)) ≤ −W3 (ku(t)k). Then solutions of Eq. (5.14) are bounded and ultimate bounded. Exercise 21. (1) (2) (3) (4)

Prove Theorem 5.7. Prove Lemma 5.4. Prove Theorem 5.8. In the proof of Theorem 5.10, show that the Poincar´e map P defined in (5.16) is continuous. (5) In the proof of Theorem 5.10, verify that as functions on · ∈ [−r, 0], h i {U (T + ·, 0)φ(0) : φ ∈ H} = {U (T + ·, k) U (k, 0)φ(0) : φ ∈ H}

given in (5.18) is equicontinuous. (6) Prove Theorem 5.11. (7) Prove Theorem 5.12. 5.1.3

Nonlinear Equations With Infinite Delay

Consider the following nonlinear differential equation with infinite delay, u0 (t) + A(t)u(t) = f (t, u(t), ut ), t > 0, u(s) = φ(s), s ≤ 0, (5.22) in a general Banach space X, where ut (s) = u(t + s), s ≤ 0. The materials presented here are based on [Liu (62); Liu Naito and Minh (63)]. Similar to equations with finite delay, we now need to consider functions defined on (−∞, 0], where in general only a seminorm is available. In this regard, an approach using axioms for seminormed abstract spaces can be made, see Henriquez [Henriquez (45)] and Hale and Kato [Hale and Kato (41)]. Here, we will approach in a simpler way by considering functions in the continuous functions space C((−∞, 0], X) against a fixed function g. These spaces are called “weighted” (or “friendly” in some literature) phase space and are denoted by Cg . To define such a phase space Cg for Eq. (5.22), we have Lemma 5.5. There exists an integer K0 > 1 such that 1 ( )K0 −1 M0 < 1, 2

(5.23)

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where M0 = supt∈[0,T ] kU (t, 0)k is finite. Next, let w0 = T /K0 , then there exists a function g on (−∞, 0] such that g(0) = 1, g(−∞) = ∞, g is decreasing on (−∞, 0], and for d ≥ w0 one has 1 g(s) ≤ . (5.24) sup g(s − d) 2 s≤0 Proof. Such a function g exists, e.g., we can take g(s) = e−as where a > 0 is such that eaw0 ≥ 2. For the function g given in Lemma 5.5, define the continuous functions space o n kφ(s)k =0 . (5.25) Cg = φ : φ ∈ C((−∞, 0], X) and lim s→−∞ g(s) Then Cg coupled with the norm kφ(s)k |φ|g = sup , φ ∈ Cg , (5.26) s≤0 g(s) is a Banach space, see an exercise. Similar to the study of equations with finite delay, we have Theorem 5.13. Let the assumptions 5.1 and 5.2 be satisfied and let φ ∈ Cg . Then there exists a constant α > 0 such that Eq. (5.22) has a unique (mild) solution u : (−∞, α] → X satisfying u0 = φ (i.e., u(s) = φ(s), s ≤ 0), and Z t U (t, h)f (h, u(h), uh )dh, t ∈ [0, α]. (5.27) u(t) = U (t, 0)φ(0) + 0

To derive periodic solutions, we assume that solutions exist on [0, ∞). We will write u = u(·, φ) for the unique solution with the initial function φ. Now, for equations with infinite delay, the Poincar´e operator will map an element φ of C((−∞, 0], X) to an element of C((−∞, 0], X), given by or, for s ∈ (−∞, 0],

P φ = uT (·, φ), φ ∈ C((−∞, 0], X),

(5.28)

(P φ)(s) = uT (s, φ) = u(T + s, φ)

=

 Z T +s    U (T + s, 0)φ(0) + U (T + s, h)f (h, u(h), uh )dh,    0

T + s > 0,

     

φ(T + s), T + s ≤ 0.

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Similar to Lemma 5.4, we have Lemma 5.6. Let the assumptions 5.1 and 5.2 be satisfied. (a). If u(t) is a solution of Eq. (5.22), then so is u(t + T ), t ≥ 0. (b). Equation (5.22) has a T -periodic solution if and only if the Poincar´e map P : C((−∞, 0], X) → C((−∞, 0], X) defined in (5.28) has a fixed point. Definitions concerning boundedness and ultimate boundedness given in the previous sections can be revised as follows to suit Eq. (5.22). Definition 5.6. The solutions of Eq. (5.22) are said to be bounded if for each B1 > 0 there is a B2 = B2 (B1 ) > 0 such that {|φ|g ≤ B1 , t ≥ 0} implies ku(t, φ)k < B2 . Definition 5.7. The solutions of Eq. (5.22) are said to be ultimate bounded if there is a bound B > 0 such that for each B3 > 0, there is a K = K(B, B3 ) > 0 such that {|φ|g ≤ B3 , t ≥ K} implies ku(t, φ)k < B. For the relationship between boundedness and ultimate boundedness, we also have the following result. Theorem 5.14. Assume that f (t, u, v) in Eq. (5.22) is continuous and is Lipschitzian in u and in v. If the solutions of Eq. (5.22) are ultimately bounded, then they are also bounded. For equations without delay and with finite delay, we obtained periodic solutions because we were able to prove the compactness of the Poincar´e operators for those equations, therefore compact sets could be constructed and Horn’s fixed point theorem could be applied to derive fixed points and hence periodic solutions. Now, for Eq. (5.22) with infinite delay, the geometry of the Poincar´e operator defined in (5.28) is that the function φ defined from 0 all the way to the left (−∞) is now mapped to a function from T (> 0) all the way to the left. Thus, under the Poincar´e operator, the history of the initial function φ on (−∞, 0] is carried over to become the part of P φ from 0 all the way to the left. Now, to obtain the compactness of the Poincar´e operator, we need to use the Ascoli-Arzela theorem and verify that the solution set (for the initial functions in a bounded set) is precompact for every s ≤ T . This is impossible now because when s ≤ 0, the solution set is the same as the initial functions, which may be arbitrary, or “bad”, i.e., noncompact. That is, it is possible that under the Poincar´e operator P defined in (5.28), a bounded set gets mapped into a noncompact set. Therefore,

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the Poincar´e operator P defined in (5.28) is not compact. Hence, all the fixed point theorems requiring compactness, such as Browder’s, Horn’s, Schauder’s, and Schauder-Tychonov’s, are not applicable now for equations with infinite delay in general Banach spaces. This causes the major difficulty for the study of periodic solutions of infinite delay equations in general Banach spaces. To overcome this difficulty, one way is to use Kuratowski’s measure of non-compactness for condensing operators to get fixed points and hence periodic solutions, where by the name of “non-compactness”, the compactness requirement is removed. Let’s introduce these notions. Definition 5.8. The Kuratowski’s measure of non-compactness (or the α measure) for a bounded set H of a Banach space Y is defined as n o α(H) = inf d > 0 : H has a finite cover of diameter < d . (5.29) We need to use the following basic properties of the α measure here.

Lemma 5.7. [Lakshmikantham and Leela (52)] Let A and B be bounded sets of a Banach space Y . Then (1) (2) (3) (4) (5) (6)

α(A) ≤ dia(A). (dia(A) = sup{|x − y|Y : x, y ∈ A}.) α(A) = 0 if and only if A is precompact. α(λA) = |λ|α(A), λ ∈ <. (λA = {λx : x ∈ A}) α(A ∪ B) = max{α(A), α(B)}. α(A + B) ≤ α(A) + α(B). (A + B = {x + y : x ∈ A, y ∈ B}) α(A) ≤ α(B) if A ⊆ B.

Lemma 5.8. Let A with norm | · |A and C with norm | · |C be bounded. If there is a surjective map Q : C → A such that for any c, d ∈ C one has |Q(c) − Q(d)|A ≤ |c − d|C , then α(A) ≤ α(C). Proof. For any ε > 0, there exist bounded sets Gi ⊆ C, i = 1, ..., m, such that i dia(Gi ) ≤ α(C) + ε, C = ∪m (5.30) i=1 G . m i Now, Q is surjective, so that A = ∪i=1 Q(G ). And for a, b ∈ Q(Gi ) we may assume that a = Q(c), b = Q(d) for some c, d ∈ Gi . Thus |a − b|A = |Q(c) − Q(d)|A ≤ |c − d|C ≤ dia(Gi ) ≤ α(C) + ε. (5.31) i This implies dia(Q(G )) ≤ α(C) + ε, and hence from Lemma 5.7 (i), α(Q(Gi )) ≤ dia(Q(Gi )) ≤ α(C) + ε. Therefore Lemma 5.7 (iv) implies that α(A) ≤ α(C) + ε. Since ε > 0 is arbitrary, the result is true.

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Definition 5.9. An operator P is said to be a condensing operator on a Banach space Y if P is continuous and takes a bounded set into a bounded set, and α(P (B)) < α(B) for every bounded set B of Y with α(B) > 0, (where α(·) is the α measure). For condensing operators, the following asymptotic fixed point theorem from Hale and Lunel [Hale and Lunel (42)] is very useful. It is similar to Horn’s and Browder’s asymptotic fixed point theorems but doesn’t require compactness. Theorem 5.15. (Hale and Lunel [Hale and Lunel (42)]) Suppose S0 ⊆ S1 ⊆ S2 are convex bounded subsets of a Banach space Y , S0 and S2 are closed, and S1 is open in S2 , and suppose P : S2 → Y is (S2 )-condensing in the following sense: if U and P (U ) are contained in S2 and α(U ) > 0, then α(P (U )) < α(U ). If P j (S1 ) ⊆ S2 , j ≥ 0, and, for any compact set H ⊆ S1 , there is a number N (H) such that P k (H) ⊆ S0 , k ≥ N (H), then P has a fixed point. Based on Theorem 5.15, we have Theorem 5.16. Suppose S0 ⊆ S1 ⊆ S2 are convex bounded subsets of a Banach space Y , S0 and S2 are closed, and S1 is open in S2 , and suppose P is a condensing operator in Y . If P j (S1 ) ⊆ S2 , j ≥ 0, and there is a number N (S1 ) such that P k (S1 ) ⊆ S0 , k ≥ N (S1 ), then P has a fixed point. The advantage of these is that the compactness requirement is removed, as long as the operator shrinks sets under α measure. We will apply the fixed point theorem 5.16 to Eq. (5.22) with infinite delay in Cg and derive periodic solutions using ultimate boundedness. For the Banach space Cg defined above, we have Lemma 5.9. Let u be a continuous function on (−∞, T ] such that |ut |g is finite for every t ∈ [0, T ]. Then for any 0 ≤ h < r ≤ T with r − h ≥ w0 (w0 is from Lemma 5.5), one has |ur |g ≤ max

n

sup ku(s)k,

s∈[h,r]

o 1 |uh |g . 2

(5.32)

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Proof.

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We have

|ur |g = sup s≤0

= sup l≤r

kur (s)k ku(r + s)k = sup g(s) g(s) s≤0 ku(l)k g(l − r)

≤ max

n

= max

n

≤ max

n

≤ max

n

(r + s = l)

sup ku(l)k

ku(l)k o 1 , sup g(l − r) l≤h g(l − r)

sup ku(l)k

ku(l)k g(l − h) o 1 , sup g(l − r) l≤h g(l − h) g(l − r)

l∈[h,r]

l∈[h,r]

sup ku(l)k, sup s≤0

l∈[h,r]

sup ku(s)k,

s∈[h,r]

o g(s) ku(h + s)k (l − h = s) g(s) g(s − (r − h))

o 1 |uh |g 2

by using Lemma 5.5.

(5.33)

To estimate the solutions, we have Lemma 5.10. Let the assumptions 5.1 and 5.2 be satisfied and let u and y be two solutions of Eq. (5.22) (with initial functions u0 and y0 respectively) on (−∞, L], L > 0. Then for t ∈ [0, L], |ut − yt |g ≤ K1 |u0 − y0 |g eK2 t ,

(5.34)

where K1 and K2 are some constants. Proof.

Similar to the proof of Lemma 5.9, we have, for t ∈ [0, L], o n (5.35) |ut − yt |g ≤ max sup ku(s) − y(s)k, |u0 − y0 |g . s∈[0,t]

Next, using Lipschitz conditions, we may assume that kf (h, u(h), yh ) − f (h, y(h), yh )k ≤ k0 ku(h) − y(h)k,

(5.36)

kf (h, u(h), uh) − f (h, u(h), yh )k ≤ k1 |uh − yh |g ,

(5.37)

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for some constants k0 and k1 . Let M0 = supt∈[0,T ] kU (t, 0)k, M1 = sup0≤h≤s≤T kU (s, h)k, which are finite. Then for s ∈ [0, t], ku(s) − y(s)k = kU (s, 0)(u(0) − y(0)) +

Z

s 0

h i U (s, h) f (h, u(h), uh ) − f (h, y(h), yh ) dhk

= kU (s, 0)(u(0) − y(0)) +

+

Z Z

s 0 s 0

h i U (s, h) f (h, u(h), uh ) − f (h, u(h), yh ) dhk h i U (s, h) f (h, u(h), yh ) − f (h, y(h), yh ) dhk

≤ M0 ku(0) − y(0)k + +

Z

Z

s 0

M1 k0 ku(h) − y(h)kdh

s 0

M1 k1 |uh − yh |g dh

≤ M0 |u0 − y0 |g +

Z

t 0

M1 (k0 + k1 )|uh − yh |g dh.

(5.38)

Thus, from (5.35), we have |ut − yt |g ≤ sup ku(s) − y(s)k + |u0 − y0 |g s∈[0,t]

≤ (M0 + 1)|u0 − y0 |g +

Z

t 0

M1 (k0 + k1 )|uh − yh |g dh.

Now, the Gronwall’s inequality implies (5.34).

An immediate consequence of Lemma 5.10 is the following local boundedness property of the solutions. Theorem 5.17. Let the assumptions 5.1 and 5.2 be satisfied and let D ⊂ Cg be bounded. Then for any L > 0, solutions of Eq. (5.22) with initial functions in D are bounded on [0, L]. That is, there exists a constant E = E(D, L) > 0 such that if u(·) = u(·, φ) with φ ∈ D, then ku(t)k ≤ E for t ∈ [0, L].

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Proof. Let y = y(φ0 ) be a fixed solution with φ0 ∈ D. Then |uL |g ≤ |uL − yL |g + |yL |g , and hence Lemma 5.10 implies that {|uL (φ)|g : φ ∈ D} is bounded. Therefore the result is true by using the definition of the norm in Cg . Next, for D ⊂ Cg and u(φ) the unique solution with u0 (φ) = φ, we define Wl (D) = {ul (φ) : φ ∈ D} and W[h,r] (D) = {u[h,r](φ) : φ ∈ D}, where u[h,r] means the restriction of u on [h, r]. Lemma 5.11. Let the assumptions 5.1 and 5.2 be satisfied. If D ⊂ Cg is bounded, then W[0,T ] (D) ⊂ C([0, T ], X) is bounded and Wr (D) ⊂ Cg is bounded for each r ∈ [0, T ]. And for any 0 ≤ h < r ≤ T with r − h ≥ w0 (w0 is from Lemma 5.5), one has n o 1 α(Wr (D)) ≤ max α(W[h,r] (D)), α(Wh (D)) . (5.39) 2 Proof. First, Theorem 5.17 implies that W[0,T ] (D) ⊂ C([0, T ], X) is bounded. This result and Lemma 5.9 (with h = 0) imply that for each r ∈ [0, T ], Wr (D) is bounded in Cg . Now, for any ε > 0, there exist bounded sets P i ⊆ W[h,r] (D), i = 1, ..., m, and bounded sets Qj ⊆ Wh (D), j = 1, ..., n, such that i dia(P i ) ≤ α(W[h,r] (D)) + ε, W[h,r] (D) = ∪m i=1 P ,

(5.40)

dia(Qj ) ≤ α(Wh (D)) + 2ε,

(5.41)

Wh (D) = ∪nj=1 Qj .

Put

Then we have

For each

Yri,j ,

o n Yri,j = ur ∈ Wr (D) : u[h,r] ∈ P i , uh ∈ Qj . i,j n Wr (D) = ∪m i=1 ∪j=1 Yr .

if ur , wr ∈

(5.42)

(5.43)

Yri,j ,

then from the proof of Lemma 5.9, o 1 |ur − wr |g ≤ max sup ku(s) − w(s)k, |uh − wh |g 2 s∈[h,r] n

o n 1 ≤ max dia(P i ), dia(Qj ) 2 n o 1 ≤ max α(W[h,r] (D)) + ε, (α(Wh (D)) + 2ε) 2 n o 1 = max α(W[h,r] (D)), α(Wh (D)) + ε. (5.44) 2

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This implies, using Lemma 5.7 (i), that n o 1 α(Yri,j ) ≤ dia(Yri,j ) ≤ max α(W[h,r] (D)), α(Wh (D)) + ε. (5.45) 2 Then Lemma 5.7 (iv) implies that n o 1 α(Wr (D)) ≤ max α(W[h,r] (D)), α(Wh (D)) + ε. (5.46) 2 Since ε > 0 is arbitrary, the result is true. By using the Ascoli-Arzela theorem, we have the following result. The idea of the proof is similar to the case for equations with finite delay. Lemma 5.12. Let the assumptions 5.1 and 5.2 be satisfied and let D ⊂ Cg be bounded. Then α(W[l,r] (D)) = 0 for any 0 < l < r ≤ T . Proof. By Lemma 5.7 (ii), we need to prove that the Ascoli-Arzela theorem can be applied to the bounded set E = W[l,r] (D) ⊂ C([l, r], X). Note that a function in E can be expressed as, for s ∈ [l, r], Z s u(s, φ) = U (s, 0)φ(0) + U (s, h)f (h, u(h), uh )dh, φ ∈ D. (5.47) 0

Since l > 0, there is k > 0 such that s > k for s ∈ [l, r]. For s ∈ [l, r], one has U (s, 0)φ(0) = U (s, k)U (k, 0)φ(0),

φ ∈ D.

(5.48)

Fix η ∈ (0, 1). Then from Lemma 5.3 (i), U (k, 0) : X → Xη is bounded. Next the embedding Xη → X is compact, thus {U (k, 0)φ(0) : φ ∈ D} is precompact in X since {φ(0) : φ ∈ D} is bounded in X. Therefore, the closure of {U (k, 0)φ(0) : φ ∈ D} is compact in X. Now, one can verify that as functions on · ∈ [l, r], h i {U (·, 0)φ(0) : φ ∈ D} = {U (·, k) U (k, 0)φ(0) : φ ∈ D} (5.49) is equicontinuous. Next, from Lemma 5.3 (ii), for 0 ≤ γ < 1, there is a constant C(γ), such that for s1 , s2 ∈ [l, r], Z s1 Z s2 U (s1 , h)f (h, u(h), uh )dhk U (s2 , h)f (h, u(h), uh )dh − k 0

0

≤ C(γ)|s1 − s2 |γ max kf (h, u(h), uh )k. (5.50) 0≤h≤T

By using Lemma 5.11, we see that the variables in f are bounded. Now f maps a bounded set into a bounded set, thus there exists M2 = M2 (D) > 0 such that kf (t, u(t), ut (φ))k ≤ M2 , t ∈ [0, T ], φ ∈ D.

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Therefore, as functions on · ∈ [l, r], Z · { U (·, h)f (h, u(h), uh )dh : φ ∈ D}

175

(5.51)

0

is also equicontinuous. Therefore, functions in E are equicontinuous. In the following, to check the precompactness of the functions at every point of [l, r], we fix s0 ∈ [l, r]. From the above arguments, we also know that {U (s0 , 0)φ(0) : φ ∈ D}

(5.52)

is precompact in X. Next, for φ ∈ D, we let g(t) = f (t, u(t), ut (φ)). Then Z s0 U (s0 , h)f (h, u(h), uh (φ))dh ≡ K(0, g)(s0 ) ∈ Xη 0

according to Lemma 5.3 (iii). Also note that by Lemma 5.3 (i), there are constants γ ∈ (0, 1) and M3 > 0 such that kU (s0 , h)k0,η ≤ M3 (s0 − h)−γ , 0 ≤ h < s0 . Thus Z k

s0

0

U (s0 , h)f (h, u(h), uh (φ))dhkη ≤ M3 M2 T 1−γ /(1 − γ), φ ∈ D.

Therefore {

Z

s0 0

U (s0 , h)f (h, u(h), uh (φ))dh : φ ∈ D}

(5.53)

is bounded in Xη . Then use the fact that the embedding Xη → X is compact again, we see that the set defined by (5.53) is precompact in X. Now the Ascoli-Arzela theorem implies that E is precompact. Thus by Lemma 5.7 (ii), α(W[l,r] (D)) = α(E) = 0. Next, we prove that the operator P defined in (5.28) is condensing in Cg . Theorem 5.18. Let the assumptions 5.1 and 5.2 be satisfied. Then the operator P defined in (5.28) is condensing in Cg with g given in Lemma 5.5. Proof.

From Lemma 5.10, we have |P (φ) − P (ϕ)|g = |uT (φ) − uT (ϕ)|g ≤ K1 eK2 T |φ − ϕ|g ,

(5.54)

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thus P is continuous, and takes a bounded set into a bounded set. Next, let D ⊂ Cg be bounded with α(D) > 0. By using Lemmas 5.11 and 5.12 repeatedly, we have (w0 is from Lemma 5.5) o n 1 α(P (D)) = α(WT (D)) ≤ max α(W[T −w0 ,T ] (D)), α(WT −w0 (D)) 2 1 α(WT −w0 (D)) 2 n o 1 1 max α(W[T −2w0 ,T −w0 ] (D)), α(WT −2w0 (D)) ≤ 2 2

=

1 = ( )2 α(WT −2w0 (D)) 2 n o 1 1 ≤ ( )2 max α(W[T −3w0 ,T −2w0 ] (D)), α(WT −3w0 (D)) 2 2 1 = ( )3 α(WT −3w0 (D)) 2

...... n o 1 1 ≤ ( )K0 −1 max α(W[0,T −(K0 −1)w0 ] (D)), α(D) . 2 2 Next, for · ∈ [0, T − (K0 − 1)w0 ], n o W[0,T −(K0 −1)w0 ] (D) ⊆ U (·, 0)φ(0) : φ ∈ D +

nZ

And for t ∈ [0, T − (K0 − 1)w0 ],

·

0

(5.55)

o U (·, h)f (h, u(h), uh (φ))dh : φ ∈ D .

kU (t, 0)φ(0) − U (t, 0)ϕ(0)k = kU (t, 0)(φ(0) − ϕ(0))k ≤ M0 kφ(0) − ϕ(0)k ≤ M0 |φ − ϕ|g ,

where M0 = supt∈[0,T ] kU (t, 0)k, then we have from Lemma 5.7 (iii) and Lemma 5.8 that (for · ∈ [0, T − (K0 − 1)w0 ]) α{U (·, 0)φ(0) : φ ∈ D} ≤ M0 α(D).

(5.56)

Similar to the proof in Lemma 5.12 we see that for · ∈ [0, T − (K0 − 1)w0 ], Z · (5.57) α{ U (·, h)f (h, u(h), uh (φ))dh : φ ∈ D} = 0. 0

Therefore we have from Lemma 5.7 (v) that

α(W[0,T −(K−1)w0 ] (D)) ≤ M0 α(D).

(5.58)

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Thus, from Lemma 5.5, (5.55), and (5.58), we have (note that M0 = supt∈[0,T ] kU (t, 0)k ≥ 1) n o 1 1 α(P (D)) ≤ ( )K0 −1 max M0 α(D), α(D) 2 2 1 ≤ ( )K0 −1 M0 α(D) < α(D). 2

This proves that the operator P is condensing in Cg .

(5.59)

Now, we are ready to prove the existence of periodic solutions for infinite delay differential equations in general Banach spaces. Theorem 5.19. Let assumptions 5.1 and 5.2 be satisfied. If the solutions of Eq. (5.22) are ultimate bounded, then Eq. (5.22) has a T -periodic solution. Proof. From Theorem 5.14, the solutions of Eq. (5.22) are also bounded. Let the operator P be defined in (5.28). Similar to the study of equations without delay and with finite delay, we have P m (φ) = umT (φ), φ ∈ Cg , m = 1, 2, · · · .

(5.60)

Next, let B > 0 be the bound in the definition of ultimate boundedness. Using boundedness, there is a B1 > B such that {|φ|g ≤ B, t ≥ 0} implies ku(t, φ)k < B1 . Also, there is a B2 > B1 such that {|φ|g ≤ B1 , t ≥ 0} implies ku(t, φ)k < B2 . Next, using ultimate boundedness, there is a positive integer J such that {|φ|g ≤ B1 , t ≥ JT } implies ku(t, φ)k < B. Now let S2 ≡ {φ ∈ Cg : |φ|g ≤ B2 }, W ≡ {φ ∈ Cg : |φ|g < B1 }, S1 ≡ W ∩ S2 ,

(5.61)

S0 ≡ {φ ∈ Cg : |φ|g ≤ B}, so that S0 ⊆ S1 ⊆ S2 are convex bounded subsets of Banach space Cg , S0

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and S2 are closed, and S1 is open in S2 . Next, for φ ∈ S1 and j ≥ 0, |P j φ|g = |ujT (φ)|g = sup s≤0

≤ max

n

≤ max

n

sup

≤ max

n

sup

sup s≤−jT

l≤0

l≤0

kujT (s)k ku(jT + s)k = sup g(s) g(s) s≤0

ku(jT + s)k , g(s)

ku(l)k , g(l − jT ) ku(l)k , g(l)

sup s∈[−jT,0]

sup ku(l)k

l∈[0,jT ]

sup ku(l)k

l∈[0,jT ]

ku(jT + s)k o g(s)

o

o

n o ≤ max |φ|g , B2 ≤ B2 ,

(5.62)

which implies P j (S1 ) ⊆ S2 , j ≥ 0. Now, we prove that there is a number N (S1 ) such that P k (S1 ) ⊆ S0 for k ≥ N (S1 ). To this end, we choose a positive integer m = m(B1 ) such that B 1 , ( )m < 2 B1

(5.63)

and then choose an integer N = N (S1 ) > J such that N T > mw0 and

B2 < B, g(−(N − J)T )

(5.64)

where w0 is from Lemma 5.5. Then for φ ∈ S1 and k ≥ N , |P k φ|g = |ukT (φ)|g = sup s≤0

≤ max

n

sup s≤−kT

kukT (s)k ku(kT + s)k = sup g(s) g(s) s≤0

ku(kT + s)k , g(s)

sup s∈[−kT,−(k−J)T ]

ku(kT + s)k , g(s)

ku(kT + s)k o . g(s) s∈[−(k−J)T,0] sup

(5.65)

For the terms in (5.65), we have sup s∈[−(k−J)T,0]

ku(kT + s)k ≤ sup ku(l)k < B, g(s) l∈[JT,kT ]

(5.66)

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≤ and sup s≤−kT

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179

ku(kT + s)k ku(l)k ≤ sup g(s) l∈[0,JT ] g(l − kT )

B2 B2 ≤ < B, g(−(k − J)T ) g(−(N − J)T )

(5.67)

ku(l)k ku(kT + s)k = sup g(s) l≤0 g(l − kT ) = sup l≤0

ku(l)k g(l) g(l) g(l − kT )

≤ |φ|g sup l≤0

≤ B1 sup l≤0

g(l) g(l − kT )

g(l) g(l − w0 ) ··· g(l − w0 ) g(l − 2w0 )

···

g(l − (m − 1)w0 ) g(l − mw0 ) . (5.68) g(l − mw0 ) g(l − kT )

Now, from Lemma 5.5, for i ≥ 0, g(l − iw0 ) g(s) sup = sup g(l − (i + 1)w ) g(s − w0 ) 0 l≤0 s≤−iw0 ≤ sup s≤0

1 g(s) ≤ . g(s − w0 ) 2

(5.69)

Thus, (5.68) becomes 1 ku(kT + s)k g(l − mw0 ) sup ≤ B1 ( )m sup g(s) 2 s≤−kT l≤0 g(l − kT )

g(l − mw0 ) g(l − mw0 ) B sup ≤ B sup = B. (5.70) B1 l≤0 g(l − N T ) l≤0 g(l − mw0 ) Therefore, (5.65) becomes |P k φ|g ≤ B, k ≥ N, (5.71) k which implies P (S1 ) ⊆ S0 , k ≥ N (S1 ). Now, Theorem 5.16 can be used to obtain a fixed point for the operator P , which, from Lemma 5.6, gives rise to a T -periodic solution of Eq. (5.22). This proves the theorem. < B1

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Theorem 5.20. Assume that there exist functions (“wedges”) Wi , i = 1, 2, 3, with Wi : [0, ∞) → [0, ∞), Wi (0) = 0, Wi strictly increasing, and W1 (t) → ∞, t → ∞. Further, assume that there exists a (Liapunov) function V : X → < (reals) such that for some constant M > 0, when u is a solution of Eq. (5.22) with ku(t)k ≥ M , then (a). W1 (ku(t)k) ≤ V (u(t)) ≤ W2 (ku(t)k), and d (b). dt V (u(t)) ≤ −W3 (ku(t)k). Then solutions of Eq. (5.22) are bounded and ultimate bounded. Exercise 22. (1) Verify that Cg coupled with the norm defined in (5.26) is a Banach space. (2) Prove Theorem 5.13. (3) Prove Lemma 5.6. (4) Prove Theorem 5.14. (5) Prove Theorem 5.16. (6) Verify (5.43). (7) In the proof of Lemma 5.12, verify that as functions on · ∈ [l, r], h i {U (·, 0)φ(0) : φ ∈ D} = {U (·, k) U (k, 0)φ(0) : φ ∈ D}

given in (5.49) is equicontinuous. (8) In the proof of Theorem 5.19, verify (5.60). (9) Prove Theorem 5.20. 5.1.4

Non-Densely Defined Equations

In the above studies of the existence of periodic solutions, the operator A(t) is assumed to be densely defined, which is the case if we look at partial differential equations in Lp spaces. However, if we use the supnorm to measure continuous functions, the corresponding operators may be non-densely defined. Example 5.1. Consider the partial differential equation  ∂2 ∂  (t, x) ∈ (0, ∞) × (0, 1),  ∂t u(t, x) = ∂x2 u(t, x) + f (t, x),    u(t, 0) = u(t, 1) = 0, t ≥ 0,      u(0, x) = Φ(x), x ∈ [0, 1].

(5.72)

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If we study Eq. (5.72) in C[0, 1] (the space of all continuous functions on [0, 1] with the sup-norm), and define Au = u00 , D(A) = {u ∈ C 2 [0, 1] : u(0) = u(1) = 0}.

(5.73)

Then the closure of D(A) is D(A) = {u ∈ C[0, 1] : u(0) = u(1) = 0} 6= C[0, 1],

(5.74)

thus A is not densely defined on C[0, 1]. In the following, we will look at u0 (t) = Au(t) + f (t, u(t), ut ), t > 0, u0 = φ ∈ C ([−r, 0] , X) , (5.75) where the linear operator A is non-densely defined and satisfies the HilleYosida condition. And we will present some results of [Ezzinbi and Liu (31)] without details. Since now the operator A is non-densely defined, the semigroup theory cannot be used. Therefore, we will apply the integrated semigroup theory. For differential equations with finite delay, the wellposedness is established in the space o n C0 = φ ∈ C ([−r, 0] , X) : φ(0) ∈ D(A) ,

so that C0 will become the base space for this setting, replacing C([−r, 0], X). The part A0 of A in D(A) is defined by o n A0 = A on D(A0 ) = x ∈ D(A) : Ax ∈ D(A) .

Then it is known that the part A0 of A generates a strongly continuous semigroup S0 (·) on D(A). Following [Liu (61)], we assume T > r and we need to verify the compactness of the Poincar´e operator P : C0 → C0 defined by P φ = uT (·, φ),

φ ∈ C0 .

(5.76)

Assumption 5.3. Let T > 0 be a constant. The function f is continuous in all its variables, T -periodic in the first variable t and uniformly Lipschitzian in other variables. Assumption 5.4. The semigroup (S0 (t))t≥0 is compact on D(A). That means for each t > 0, the operator S0 (t) is compact on D(A).

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The definitions on boundedness and ultimate boundedness are the same as those for finite delay equations studied before. With some conditions, the approach of [Liu (61)] can be modified to construct some sets so that Horn’s fixed point theorem can be applied to derive fixed points for the Poincar´e operator defined in (5.76), and hence periodic solutions. We state these results as follows. Theorem 5.21. Assume that f (t, u, v) in Eq. (5.75) is continuous and is Lipschitzian in u and in v. If the solutions of Eq. (5.75) are ultimately bounded, then they are also bounded. Theorem 5.22. Let the assumptions 5.3 and 5.4 be satisfied and let T > r. If the solutions of Eq. (5.75) are bounded, then the Poincar´e operator P φ = uT (·, φ) on C0 defined in (5.76) is compact. Theorem 5.23. Let the assumptions 5.3 and 5.4 be satisfied and let T > r. If the solutions of Eq. (5.75) are ultimately bounded, then Eq. (5.75) has a T -periodic solution. Theorem 5.24. Assume that there exist functions (“wedges”) Wi , i = 1, 2, 3, with Wi : [0, ∞) → [0, ∞), Wi (0) = 0, Wi strictly increasing, and W1 (t) → ∞, t → ∞. Further, assume that there exists a (Liapunov) function V : X → < (reals) such that for some constant M > 0, when u is a solution of Eq. (5.75) with ku(t)k ≥ M , then (a). W1 (ku(t)k) ≤ V (u(t)) ≤ W2 (ku(t)k), and d V (u(t)) ≤ −W3 (ku(t)k). (b). dt Then solutions of Eq. (5.75) are bounded and ultimate bounded. Exercise 23. (1) In Example 5.1, verify that D(A) = {u ∈ C[0, 1] : u(0) = u(1) = 0} 6= C[0, 1]. (2) Study the wellposedness of Eq. (5.75) in n o C0 = φ ∈ C ([−r, 0] , X) : φ(0) ∈ D(A) .

(3) Prove that the part A0 of A generates a strongly continuous semigroup T0 (·) on D(A). (4) Prove Theorem 5.21. (5) Prove Theorem 5.22. (6) Prove Theorem 5.23. (7) Prove Theorem 5.24.

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5.2

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Evolution Semigroups and Almost Periodic Solutions

The studies of the previous sections can be extended to almost periodic solutions of nonlinear equations. The first question we are faced with is how to associate to a given nonlinear evolution equation evolution semigroups in suitable function spaces. The next one is to find common fixed points of these semigroups in chosen function spaces. It turns out that using evolution semigroups we can not only give simple proofs of some results in the finite dimensional case, but also extend them easily to the infinite dimensional case. 5.2.1

Evolution Semigroups

In this subsection we are mainly concerned with the existence of almost periodic solutions of evolution equations of the form dx = Ax + f (t, x, xt ) (5.77) dt where A is the infinitesimal generator of a C0 -semigroup (T (t))t≥0 and f is a continuous operator from R × X × C to X. Note that our method used in this subsection applies to nonautonomous equations with almost periodic coefficients, not restricted to periodic or autonomous equations as in the previous sections. Throughout this section we will denote by C = BU C((−∞, 0], X) the space of all uniformly continuous and bounded functions from (−∞, 0] to X, and by xt the map x(t + θ) = xt (θ), θ ∈ (−∞, 0], where x(·) is defined on (−∞, a] for some a > 0. In this subsection we will deal with evolution equations of the form dx = Ax + f (t, x) , x ∈ X (5.78) dt where X is a Banach space, A is the infinitesimal generator of a C0 semigroup of linear operators (S(t))t≥0 of type ω, i.e. kS(t)x − S(t)yk ≤ eωt kx − yk, ∀ t ≥ 0, x, y ∈ X , and f is a continuous operator from R × X to X. Hereafter, recall that by a mild solution x(t), t ∈ [s, τ ] of equation (5.78) we mean a continuous solution of the integral equation Z t x(t) = S(t − s)x + S(t − ξ)f (ξ, x(ξ)dξ, ∀s ≤ t ≤ τ. (5.79) s

Definition 5.10. (condition H4). Equation (5.78) is said to satisfy condition H4 if

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(1) A is the infinitesimal generator of a linear semigroup (S(t))t≥0 of type ω in X, (2) f is a continuous operator from R × X to X, (3) There is a constant γ such that for every fixed t ∈ R, the operator (−f (t, ·) + γI) is accretive in X. The following condition will be used frequently: Definition 5.11. (condition H5). Equation (5.78) is said to satisfy condition H5 if for every u ∈ AP (X) the function f (·, u(·)) belongs to AP (X) and the operator f∗ taking u into f (·, u(·)) is continuous. The main point of our study is to associate with equation (5.78) an evolution semigroup which plays a role similar to that of the monodromy operator for equations with periodic coefficients. Hereafter we will denote by U (t, s), t ≥ s, the evolution operator corresponding to equation (5.78) which satisfies the assumptions of Theorem A.30, i.e. U (t, s)x is the unique solution of Eq. (5.79). Proposition 5.1. Let the conditions H4 and H5 be satisfied. Then with Eq. (5.78) one can associate an evolution semigroup (T h )h≥0 acting on AP (X), defined as [T h v](t) = U (t, t − h)v(t − h), ∀h ≥ 0, t ∈ R, v ∈ AP (X). Moreover, this semigroup has the following properties: (1) T h , h ≥ 0 is strongly continuous, and Z h T hu = S h u + S h−ξ f∗ (T ξ u)dξ, ∀h ≥ 0, u ∈ AP (X), 0

h

(2)

where (S u)(t) = S(h)u(t − h), ∀h ≥ 0, t ∈ R, u ∈ AP (X). kT h u − T h vk ≤ e(ω+γ)h ku − vk, ∀h ≥ 0, u, v ∈ AP (X).

Proof.

We first look at the solutions to the equation Z t S t−ξ f∗ (w(ξ))dξ ∀z ∈ AP (X), t ≥ a ∈ R. w(t) = S t−a z +

(5.80)

a

It may be noted that (S h )h≥0 is a strongly continuous semigroup of linear oparators in AP (X) of type ω. Furthermore, for λ > 0, λγ < 1 and u, v ∈

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AP (X), from the accretiveness of the operators −f (t, ·) + γI we get (1 − λγ)kx − yk = (1 − λγ) sup ku(t) − v(t)k t

= sup(1 − λγ)ku(t) − v(t)k t

≤ sup ku(t) − v(t) − λ[f (t, u(t)) − f (t, v(t))]k t

= ku − v − λ(f∗ u − f∗ v)k.

(5.81)

This shows that (−f∗ + γI) is accretive. In virtue of Theorem A.30 there exists a semigroup (T h )h≥0 such that h

h

T u=S u+

Z

h 0

S h−ξ f∗ T ξ udξ,

kT h u − T h vk ≤ e(ω+γ)h ku − vk, ∀h ≥ 0, u, v ∈ AP (X). From this, Z

[T h u](t) = [S h u](t) +

h 0

[S h−ξ f∗ (T ξ u)](t)dξ, ∀t ∈ R.

Thus h

[T u](t) = S(h)u(t − h) + = S(h)u(t − h) + = S(h)u(t − h) +

Z

Z

Z

h

S(h − ξ)[f∗ (T ξ u](t − h + ξ)dξ

0 h 0

S(h − ξ)f (t + ξ − h, [T u ](t + ξ − h))dξ

t t−h

S(t − η)f (η, [T η−(t−h) u](η)dη.

If we denote [T t−s u](t) by x(t), we get x(t) = S(t − s)z +

Z

t s

S(t − ξ)f (ξ, x(ξ))dξ, ∀t ≥ s,

(5.82)

where z = u(s). Consequently, from the uniqueness of mild solutions Equation (5.78) we get [T t−s u](t) = x(t) = U (t, s)u(s) and [T h u](t) U (t, t − h)u(t − h) for all t ≥ s, u ∈ AP (Q). This completes the proof the proposition.

of = of

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Almost periodic solutions

5.2.2.1 Almost periodic solutions of differential equations without delay The main idea underlying our approach is the following assertion. Corollary 5.1. Let all assumptions of Proposition 5.1 be satisfied. Then a mild solution x(t) of Eq. (5.77), defined on the whole real line R, is almost periodic if and only if it is a common fixed point of the evolution semigroup (T h )h≥0 defined in Proposition 5.1. Proof. Suppose that x(t), defined on the real line R, is an almost periodic mild solution of Eq. (5.78). Then from the uniqueness of mild solutions we get x(t) = U (t, t − h)x(t − h) = [T h x](t), ∀t ∈ R. This shows that x is a fixed point of T h for every h > 0. Conversely, suppose that y(·) is any common fixed point of T h , h ≥ 0. Then y(t) = [T t−s y](t) = U (t, s)y(s), ∀t ≥ s. This shows that y(·) is a mild solution of Eq. (5.78).

We now apply Corollary 5.1 to find sufficient conditions for the existence of almost periodic mild solutions of Eq. (5.78). Corollary 5.2. Let all conditions of Proposition 5.1 be satisfied. Furthermore, let ω + µ be negative and −f∗ − µI be accretive. Then there exists a unique almost periodic mild solution of Eq. (5.78). Proof. It is obvious that there exists a unique common fixed point of the semigroup (T h )h≥0 . The assertion now follows from Corollary 5.1. Remark 5.2. (1) A particular case in which we can check the accretiveness of −f∗ − µI is ω + γ < 0. In fact, this follows easily from the above estimates for ku − vk (see the estimate (5.81)). (2) It is interesting to ”compute” the infinitesimal generator of the evolution semigroup (T h )h≥0 determined by Proposition 5.1. To this purpose, let us recall the operator L which relates a mild solution u of the equation x˙ = Ax + f (t) to the forcing term f by the rule Lu = f (see

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Sections 2.1 and 2.2 for more discussion on this operator). From the proof of Proposition 5.1 it follows in particular that the infinitesimal generator G of the evolution semigroup (T h )h≥0 is −L + f∗ . (3) It can be seen that u is a mild solution of Eq. (5.78) if and only if (−L + f∗ )u = 0, so it is the fixed point of the semigroup (T h )h≥0 . (4) Let f∗ act on the function space Λ(X) ∩ AP (X). Then by the same argument as in the proof of Proposition 5.1 we can prove that the evolution semigroup (T h )h≥0 leaves Λ(X) ∩ AP (X) invariant. This will be helpful if we want to discuss the spectrum of the unique almost periodic solution in Corollary 5.2. 5.2.2.2 Almost periodic solutions of differential equations with delays In this subsection we apply the results of the previous subsection to study the existence of almost periodic mild solutions of the equation dx = Ax + f (t, x, xt ) (5.77) dt where A is defined as in the previous subsection, and f is an everywhere defined continuous mapping from R × X × C to X. Hereafter we call a continuous function x(t) defined on the real line R a mild solution of Eq. (5.77) if Z t x(t) = S(t − s)x(s) + S(t − ξ)f (ξ, x(ξ), xξ )dξ, ∀t ≥ s. s

We should emphasize that our study is concerned only with the existence of almost periodic mild solutions of Eq. (5.77), and not with all mild solutions in general. Definition 5.12. (condition H6). Equation (5.77) is said to satisfy condition H6 if the following is true:

(1) For every g ∈ AP (X) the mapping F (t, x) = f (t, x, gt ) satisfies conditions H4 and H5 with the same constant γ. (2) There exists a constant µ with ω − µ < 0 such that −(µI + F∗ ) is accretive for every g ∈ AP (X). (3) [x − y, f (t, x, φ) − f (t, y, φ0 )] ≤ γkx − yk + δkφ − φ0 k, ∀t ∈ R, x, y ∈ X, φ, φ0 ∈ C. Theorem 5.25. Let condition H6 hold. Then for δ sufficiently small (see the estimate (5.86) below), Eq. (5.77) has an almost periodic mild solution.

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Proof. First we fix a function g ∈ AP (X). In view of Proposition 5.1 we observe that the equation dx = Ax + F (t, x) dt has a unique almost periodic mild solution, where F (t, x) = f (t, x, gt ). We denote this solution by T g. Thus, we have defined an operator T acting on AP (X). We now prove that T is a strict contraction mapping. In fact, let us denote by U (t, s) and V (t, s) the Cauchy operators Z t S(t − ξ)f (ξ, U (ξ, s)x, gξ )dξ, (5.83) U (t, s)x = S(t − s)x + s Z t V (t, s)x = S(t − s)x + S(t − ξ)f (ξ, V (ξ, s)x, hξ )dξ, (5.84) s

for given g, h ∈ AP (X), x ∈ X, t ≥ s. Putting u(t) = U (t, s)x, v(t) = V (t, s)x for given s, x, from the assumptions we have [u(t) − v(t), f (t, u(t), gt − f (t, v(t), ht ] ≤ m(t, ku(t) − v(t)k), where m(t, ku(t) − v(t)k) = γku(t) − v(t)k + δkh − gk. Using this we get ku(t) − v(t)k ≤ ku(t − η) − v(t − η)k + ηm(t, ku(t) − v(t)k) Z t + kS(t − ξ)f (ξ, u(ξ), hξ ) − f (t, u(t), ht )kdξ t−η t

+

Z

t−η

kS(t − ξ)f (ξ, v(ξ), gξ ) − f (t, v(t), gt )kdξ.

Now let us fix arbitrary real numbers a ≤ b. Since the functions S(t − ξ)f (ξ, u(ξ), hξ ) and S(t − ξ)f (ξ, v(ξ), gξ ) are uniformly continuous on the set a ≤ ξ ≤ t ≤ b, for every ε > 0 there exists an η0 = η0 (ε) such that kS(t − ξ)f (ξ, u(ξ), hξ ) − f (t, u(t), ht )k < ε,

kS(t − ξ)f (ξ, v(ξ), gξ ) − f (t, v(t), gt )k < ε,

for all kt − ξk < η0 and t ≤ ξ ∈ [a, b]. Hence, denoting ku(t) − v(t)k by α(t), for η < η0 we have α(t) − eωη α(t − η) ≤ ηm(t, α(t)) + 2ηε.

(5.85)

Applying this estimate repeatedly, we get α(t) − eω(t−s) ≤

n X i=1

eω(t−ti ) m(ti , α(ti ))∆i + 2ε

n X i=1

eω(t−ti ) ∆i ,

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where t0 = s < t1 < t2 < ... < tn = t and |ti − ti−1 | = ∆i . Thus, since ε is arbitrary, and since the function m is continuous, we get Z t ω(t−s) α(t) − e α(s) ≤ eω(t−ξ) m(ξ, α(ξ))dξ s Z t eω(t−ξ) (γα(ξ) + δkh − gk)dξ. = s

Applying Gronwall’s inequality we get

e−ωs − e−ωt )δkh − gk. ω Because of the identity α(s) = ku(s) − v(s)k = kU (s, s)x − V (s, s)xk = 0, from the above estimate we obtain eγ+ω − eγ sup kU (ξ, t − 1)x − V (ξ, t − 1)xk ≤ δkh − gk. ω t−1≤ξ≤t α(t) ≤ e(γ+ω)(t−s) α(s) + eγ(t−s)+ωt (

Now let us denote by Tht , Tgt , t ≥ 0 the respective evolution semigroups corresponding to Eq. (5.83) and Eq. (5.84). Since T h and T g are defined as the unique fixed points u0 , v0 of Th1 , Tg1 , respectively, we have kT h − T gk = ku0 − v0 k = kTh1 u0 − Tg1 v0 k ≤

≤ kTh1 u0 − Tg1 u0 k + kTg1 u0 − Tg1 − v0 k

eγ+ω − eγ δkh − gk + eω−µ ku0 − v0 k ω = N δkh − gk + eω−µ kT h − T gk, ≤

where N = (eγ+ω − eγ )/ω. Finally, we have eγ (eω − 1) kT h − T gk ≤ . ω(1 − eω−µ ) Thus, if the estimate ω(1 − eω−µ ) δ< γ ω (5.86) e (e − 1) holds true, then T is a strict contraction mapping in AP (X). By virtue of the Contraction Mapping Principle T has a unique fixed point. It is easy to see that this fixed point is an almost periodic mild solution of Eq. (5.77). This completes the proof of the theorem. Remark 5.3. (1) In case ω = 0, γ = −µ we get the estimate

δ < eµ − 1 = µ + µ2 /2 + ...

which guarantees the existence of the fixed point of T .

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(2) If ω + γ < 0 , then we can choose µ = −γ, and therefore we get the accretiveness condition on −(F∗ + µI). However, in general, the condition ω + γ < 0 is a very strong restriction on the coefficients of Eq. (5.77), if f depends explicitly on t. 5.2.2.3 Examples In applications one frequently encounters functions f from R × X × C → X of the form f (t, x, gt ) = F (t, x) + G(t, gt ), ∀t ∈ R, x ∈ X, gt ∈ C,

where F satisfies condition ii) of Definition 5.12 and G(t, y) is Lipschitz continuous with respect to y ∈ C, i.e. kG(t, y) − G(t, z)k ≤ δky − zk, ∀t ∈ R, y, z ∈ C

for some positive constant δ. In order to describe a concrete example we consider a bounded domain Ω in Rn with smooth boundary ∂Ω and suppose that X A(x, D)u = aα (x)Dα u |α|≤2m

is a strongly elliptic differential operator in Ω. Then, defining the operator Au = A(x, D)u, ∀u ∈ D(A) = W 2m,2 (Ω) ∩ W0m,2 (Ω)

we know from Theorem 3.6 in [Pazy (90)] that the operator −A is the infinitesimal generator of an analytic semigroup of contractions on L2 (Ω). Now let f, g : R × Ω × R → R be Lipschitz continuous and define the operators F (t, w)(x) = f (t, x, w(x)) and G(t, w)(x) = g(t, x, w(x)) where t ∈ R, x ∈ Ω and w ∈ L2 (Ω). Then, for any positive constant r, the boundary value problem ∂u(t, x) = A(x, D)u(t, x) + f (t, x, u(t, x)) + g(t, x, u(t − r, x)) in Ω , ∂t u(t, x) = 0 on ∂Ω fits into the abstract setting of Eq. (5.77).

5.3 5.3.1

Comments and Further Reading Guide Further Reading Guide

For Eq. (5.22), the idea of deriving periodic solutions using boundedness has been recently extended to general fading memory phase spaces with

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axioms in [Ezzinbi, Liu and Minh (32)]. So that in some sense, the study along this line is getting complete. A phase space for Eq. (5.22) is called a fading memory space if it is a Banach space (Γ, k · kΓ ) consisting of functions from (−∞, 0] to X that satisfy the following axioms ([Hale and Kato (41); Hino and Murakami (47)]): (A1). There exist a positive constant H and locally bounded non-negative continuous functions K(·) and M (·) on [0, ∞) with the property that if u : (−∞, a) → X is continuous on [σ, a) with uσ ∈ Γ for some σ < a, then for all t ∈ [σ, a), (i). ut ∈ Γ, (ii). ut is continuous in t (with respect to k · kΓ ), (iii). Hkx(t)k ≤ kxt kΓ ≤ K(t − σ) supσ≤s≤t kx(s)k + M (t − σ)kxσ kΓ .

(A2). If {φk }, φk ∈ Γ, converges to φ uniformly on any compact set in (−∞, 0] and if {φk } is a Cauchy sequence in Γ, then φ ∈ B and φk → φ in Γ, k → ∞. A fading memory space is called a uniform fading memory space if it satisfies (A1) and (A2) with K(·) ≡ K1 (a constant) and M (t) → 0 as t → ∞. In [Ezzinbi, Liu and Minh (32)], the Poincar´e operator is shown to be condensing in Γ under the condition that M (0) < 1, which simplifies a condition in [Henriquez (45)] of the form 1 (5.87) inf M (T − σ)[ K(σ) sup kT (t)k + M (σ)] < 1. 0<σ
Comments

Most results here are obtained under the condition that the nonlinear function f is Lipschitzian in variables other than t. If f is an arbitrary nonlinear function, then we need to also assume that f maps a bounded set into a bounded set and require that the solutions are also bounded (in addition to being ultimate boundeded) in order to carry the proofs.

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Appendix

A.1 Lipschitz Operators Let X and Y be given Banach spaces over the same field R, C. An operator A : X → Y is called Lipschitz continuous if there is a positive constant L such that kAx − AykY ≤ Lkx − ykX , ∀x, y ∈ X. For a Lipschitz continuous operator A the following kAk :=

sup x,y∈X, x6=y

kAx − Ayk/kx − yk

is finite and is called the Lipschitz constant of A. The set of all Lipschitz continuous operators from X to Y is denoted by Lip(X; Y) and Lip(X; X) = Lip(X) for short. A member A ∈ Lip(X) is said to be invertible if there is a B ∈ Lip(X) such that A · B = B · A = I. B is called the inverse of A and is denoted by A−1 Theorem A.26. (Lipschitz Inverse Mapping) Let X be a Banach space, A is an invertible member of Lip(X) and B is a member of Lip(X) such that kBk · kA−1 k < 1. Then A + B is invertible in Lip(X) and k(A + B)−1 k ≤ kA−1 k(1 − kBk · kA−1 k)−1 . Proof. We first prove the following assertion: If A ∈ Lip(X) such that kAk < 1. Then (I − A) is invertible in Lip(X) and k(I − A)−1 k ≤ (1 − kAk)−1 .

(A.1)

In fact, for x, y ∈ X k(I − A)x − (I − A)yk ≥ kx − yk − kAx − Ayk ≥ (1 − kAk)kx − yk. 193

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Thus I − A is injective. If z, w ∈ R(I − A), then

k(I − A)−1 z − (I − A)−1 wk ≤ (1 − kAk)−1 kz − wk.

For x ∈ X by induction we can prove that

kBn+1 x − Bn xk ≤ kAkn kAxk, ∀n = 0, 1, 2, ...

where by induction we define B0 := I, Bn := I + ABn−1 , ∀n = 1, 2.... Indeed, this holds true for n = 0, so if we assume it to be true for n − k, then kBk+1 x − Bk xk = kABk x − ABk−1 xk

≤ kAk · kBk x − Bk−1 xk ≤ kAk · kAkk−1 kAxk,

so the assertion follows by induction. For any positive integer p,

p−1

X

kBn+p x − Bn xk = (Bn+k+1 x − Bn+k x)

k=0

≤

≤

p−1 X

k=0

p−1 X

k=0

k(Bn+k+1 x − Bn+k x)k

kAkn+k kAxk ≤ kAn kkAxk(1 − kAk)−1 .

Since kAk < 1 and X is a Banach space, Cx = limm→∞ Bm x exists for all x ∈ X and kCx − Bn xk = lim kBn+p x − Bn xk ≤ kAkn kAxk(1 − kAk)−1 . p→∞

Since A is continuous, Cx = lim Bn x = lim (I − ABn−1 )x = x + ACx. n→∞

n→∞

This shows that C = I +AC, so C is a right inverse of I −A, i.e. (I −A)C = I. Finally, this shows the surjectiveness of I − A, proving the assertion that I − A is invertible in Lip(X). We are now in a position to prove the theorem. In fact, we have (A + B) = (I + BA−1 )A and kBA−1 k ≤ kBk · kA−1 k < 1. By the above assertion, (I +BA−1 )−1 exists as an element of Lip(X). Hence (A+B)−1 = A−1 (I + BA−1 )−1 . and k(I + BA−1 )−1 k ≤ (1 − kBk · kA−1 k)−1 .

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A modification of the above theory for Lipschitz continuous operators from a Banach space X to another Banach space Y can be easily made. For instance, the following is true: Theorem A.27. Let A be an invertible member of Lip(X, Y). Then for sufficiently small positive k, the operator A + B is an invertible member of Lip(X, Y) if kBk < k. Proof. Set C = A−1 (A + B) − I. Then C is a member of Lip(X), and for all x, y ∈ X, kCx − CykX = k(A−1 (A + B) − A−1 A)x − (A−1 (A + B) − A−1 A)yk ≤ kA−1 k · kBk · kx − yk.

Thus for sufficiently small positive k, I + C is invertible, so is A + B.

A.2 Fixed Point Theorems The following fixed point theorems can be found in Smart [Smart (99)], Burton [Burton (18)], and Hale and Lunel [Hale and Lunel (42)]. Definition. A mapping (operator) P on a metric space (X, ρ) is called a contraction mapping if there is an r ∈ (0, 1) such that ρ(P x, P y) ≤ rρ(x, y).

Theorem (Contraction mapping principle). Let P be a contraction mapping on a complete metric space X, then there is a unique x ∈ X with P x = x. Moreover, x = limn→∞ xn , where x0 is any element of X and xj+1 = P xj , j = 0, 1, · · · . Proof. Now, for some 0 < r < 1, we have ρ(P y, P z) ≤ rρ(y, z) when y, z ∈ X. Let x0 be any element of X and define xj+1 = P xj , j = 0, 1, · · · . Then x1 = P x0 , x2 = P x1 = P 2 x0 , · · · , xj = P xj−1 = · · · = P j x0 , j =

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1, 2, · · · . Thus, for m > n,

ρ(xn , xm ) = ρ(P n x0 , P m x0 ) ≤ rρ(P n−1 x0 , P m−1 x0 ) .. . ≤ rn ρ(x0 , P m−n x0 ) = rn ρ(x0 , xm−n ) h i ≤ rn ρ(x0 , x1 ) + ρ(x1 , x2 ) + · · · + ρ(xm−n−1 , xm−n ) h i ≤ rn ρ(x0 , x1 ) + rρ(x0 , x1 ) + · · · + rm−n−1 ρ(x0 , x1 ) h i = rn ρ(x0 , x1 ) 1 + r + · · · + rm−n−1 ≤ rn ρ(x0 , x1 )

1 . 1−r

(A.2)

As 0 < r < 1, the right-hand side goes to zero when n → ∞. Thus {xn } is a Cauchy sequence, and hence has a limit x ∈ X because X is a complete metric space. Now, it is easily seen that P is continuous, therefore (A.3) P x = P lim xn = lim P xn = lim xn+1 = x, n→∞

n→∞

n→∞

and x is a fixed point of P . If y is also a fixed point of P , then ρ(x, y) = ρ(P x, P y) ≤ rρ(x, y),

(A.4)

and, as 0 < r < 1, we must have ρ(x, y) = 0, which implies x = y. This completes the proof. Theorem (Brouwer’s fixed point theorem). Let B ⊂
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Definition. Let A and B be subsets of a Banach space Z. If A = B ∩ C for an open subset C of Z, then A is open relative to B. Theorem (Horn’s fixed point theorem). Let E0 ⊂ E1 ⊂ E2 be convex subsets of a Banach space Z, with E0 and E2 compact subsets and E1 open relative to E2 . Let P : E2 → Z be a continuous operator such that for some integer m, one has P j (E1 ) ⊂ E2 , 1 ≤ j ≤ m − 1,

(A.5)

P j (E1 ) ⊂ E0 , m ≤ j ≤ 2m − 1,

(A.6)

then P has a fixed point in E2 .

Theorem (Browder’s fixed point theorem). Let E0 ⊂ E1 ⊂ E2 be convex subsets of a Banach space Z, with E0 closed and E1 , E2 open. Let P : E2 → Z be a compact operator such that for some integer m, one has P j (E0 ) ⊂ E1 , 0 ≤ j ≤ m,

P m (E1 ) ⊂ E0 ,

(A.7)

(A.8)

then P has a fixed point in E2 .

Theorem (Hale and Lunel’s fixed point theorem). Suppose S0 ⊆ S1 ⊆ S2 are convex bounded subsets of a Banach space Y , S0 and S2 are closed, and S1 is open in S2 , and suppose P : S2 → Y is (S2 )-condensing in the following sense: if U and P (U ) are contained in S2 and α(U ) > 0, then α(P (U )) < α(U ). If P j (S1 ) ⊆ S2 , j ≥ 0, and, for any compact set H ⊆ S1 , there is a number N (H) such that P k (H) ⊆ S0 , k ≥ N (H), then P has a fixed point.

A.3 Invariant Subspaces Let S ⊂ H be a closed subset and PS , the orthogonal projection onto the subspace S. The operator is still a densely defined closed (possibly unbounded) linear operator in H. Definition A.13. S is said to be an invariant subspace for A if we have the inclusion A(D(A) ∩ S) ⊂ S. Example A.2. Let us mention the following classical invariant subspaces for the closed unbounded linear operator A defined into the Hilbert space H.

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1. S = N (A) = {x ∈ D(A) : Ax = 0} is an invariant subspace for A.

2. If A is a self-adjoint linear operator, then any eigenspace Sλ = N (λI − A) is an invariant for A. In fact it can be easily shown that Sλ reduces A. Theorem A.28. The equality PS APS = APS is a necessary and sufficient condition for a subspace S to be invariant for a linear operator A. Proof. Assume PS APS = APS and if x ∈ D(A) ∩ S, then x = PS x ∈ D(A) and Ax = APS x = PS APS x ∈ S. Conversely, if S is invariant for A; let x ∈ H be such that PS x ∈ D(A). Then APS x ∈ S and then PS APS x = APS x. Therefore APS ⊂ PS APS . Since D(APS ) = D(PS APS ), it turns out that APS = PS APS . Definition A.14. A closed proper subspace S of the Hilbert space H is said to reduce an operator A if PS D(A) ⊂ D(A) and both S and H S, the orthogonal complement of S, are invariant for A. Using the above Theorem , the following key result can be proved. Theorem A.29. A closed subspace S of H reduces an operator A if and only if PS A ⊂ APS . Proof.

See the proof in [Locker (64)] Theorem 4.11., p. 29.

Remark A.4. In fact the meaning of the inclusion PS A ⊂ APS is that: if x ∈ D(A) , then PS x ∈ D(A) and PS Ax = APS x . A.4 Semilinear Evolution Equations We recall in this section a result on the well posedness for semilinear equations of the form dx = Ax + Bx , x ∈ X (A.9) dt where X is a Banach space, A is the infinitesimal generator of a C0 semigroup S(t), t ≥ 0 of linear operators of type ω, i.e. kS(t)x − S(t)yk ≤ eωt kx − yk, ∀ t ≥ 0, x, y ∈ X , and B is an everywhere defined continuous operator from X to X. Hereafter, by a mild solution x(t), t ∈ [s, τ ] of equation (A.9) we mean a continuous solution of the integral equation Z t x(t) = S(t − s)x + S(t − ξ)Bx(ξ)dξ, ∀s ≤ t ≤ τ. (A.10) s

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Before proceeding we recall some notions and results which will be frequently used later on. We define the bracket [·, ·] in a Banach space Y as follows (see e.g. [Martin (67)] for more information) kx + hyk − kyk kx + hyk − kyk = inf h→+0 h>0 h h

[x, y] = lim

Definition A.15. Suppose that F is a given operator on a Banach space Y. Then (F + γI) is said to be accretive if and only if for every λ > 0 one of the following equivalent conditions is satisfied (1) (1 − λγ)kx − yk ≤ kx − y + λ(F x − F y)k, ∀x, y ∈ D(F ), (2) [x − y, F x − F y] ≥ −γkx − yk, ∀x, y ∈ D(F ). In particular, if γ = 0 , then F is said to be accretive. Remark A.5. From this definition we may conclude that (F + γI) is accretive if and only if kx − yk ≤ kx − y + λ(F x − F y)k + λγkx − yk

(A.11)

for all x, y ∈ D(F ), λ > 0, 1 ≥ λγ . Theorem A.30. Let the above conditions hold true. Then for every fixed s ∈ R and x ∈ X there exists a unique mild solution x(·) of Eq.(A.9) defined on [s, +∞). Moreover, the mild solutions of Eq.(A.9) give rise to a semigroup of nonlinear operators T (t), t ≥ 0 having the following properties: Z t i) T (t)x = S(t)x + S(t − ξ)BT (ξ)xdξ, ∀t ≥ 0, x ∈ X, (A.12) 0

ii) kT (t)x − T (t)yk ≤ e(ω+γ)t kx − yk, ∀t ≥ 0, x, y ∈ X.

(A.13)

More detailedly information on this subject can be found in [Martin (67)]. Theorem A.31. Let D be a closed and convex subset of a Hausdorff locally convex space such that 0 ∈ D, and let G be a continuous mapping of D into itself. If the implication (V = convG(V )

or

V = G(V ) ∪ {0}) =⇒ V is relatively compact

holds for every subset V of D, then G has a fixed point.

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In what follows we shall need the following definition of the proper mapping at a given point. Let X, Y be metric spaces, f : X → Y be a continuous function and let y ∈ Y . f is said to be proper at the point y provided that there exists ε > 0 such that for any compact set K ⊂ B(y, ε) the set f −1 (K) is compact, where B(y, ε) is the open ball in Y of center y and radius ε. If f −1 (K) is compact for any compact K ⊂ Y , then f is called proper. The proofs of Aronszajn type results (see [Aronszjan (9)]) are based on the following Browder-Gupta type theorem (see [G´ orniewicz (38)]) Theorem A.32. Let E be a Banach space and f : X → E be a continuous map such that the following conditions are satisfied: (i) f is proper at 0 ∈ E, (ii) for every ε > 0 there exists a continuous map fε : X → E for which we have: (a) kf (x) − fε (x)k < ε for every x ∈ X, (b) the map f˜ε : fε−1 (B(0, ε)) → B(0, ε), f˜ε (x) = fε (x) for every x ∈ fε−1 (B(0, ε)), is a homeomorphism. Then the set f −1 ({0}) is an Rδ set.

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nal of Differential Equations 208 (2005), 388-429. Y. Latushkin, F. R¨ abiger, Operator valued Fourier multipliers and stability of strongly continuous semigroups. Integral Equations Operator Theory, 51 (2005), 375-394. B.M. Levitan, V.V. Zhikov, ”Almost Periodic Functions and Differential Equations”, Moscow Univ. Publ. House 1978. English translation by Cambridge University Press 1982. J. Liu, A First Course in the Qualitative Theory of Differential Equations, Prentice Hall, New Jersey, 2003. J. Liu, Bounded and periodic solutions of semi-linear evolution equations, Dynamic Sys. & Appl., 4(1995), 341-350. J. Liu, Bounded and periodic solutions of finite delay evolution equations, Nonlinear Anal., 34(1998), 101-111. J. Liu, Periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 247(2000), 627-644. J. Liu, T. Naito, and N. Minh, Bounded and periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 286(2003), 705-712. J. Locker, ”Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators”, Amer. Math. Soc. Mathematical Surveys and Monographs, Vol. 73, (2000). A. Lunardi, ”Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Birhauser, Basel, 1995. Yu. I. Lyubich, Vu Quoc Phong, Asymptotic stability of linear differential equations on Banach spaces, Studia Math. 88 (1988), 37-42. R. Martin, ”Nonlinear operators and differential equations in Banach spaces”, Wiley-Interscience, New York 1976. J.L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J.17, (1950). 457–475. N.V. Minh, On the proof of characterisations of the exponential dichotomy, Proc. Amer. Math. Soc 127(1999), 779-782. N.V. Minh, F. R¨ abiger, R. Schnaubelt, On the exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Int. Eq. and Oper. Theorey 32(1998), 332-353. S. Murakami, T. Naito, N.V. Minh, Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces, J. Diff. Eq., 164 (2000), 240-285. Nguyen Van Minh, Ha Binh Minh, A Marssera - type theorem for almost periodic solutions of higher order delay or advance abstract functional differential equations, Abstract and Applied Analysis Vol. 2004 (2004), 881-896. R. Nagel ,(Ed) ”One-parameter Semigroups of Positive Operators”, Springer. Lec. Notes in Math. 1184(1986). T. Naito and N.V. Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Diff. Eq. 152(1999), 358376. T. Naito, N.V. Minh, R. Miyazaki, J.S. Shin, A decomposition theorem for bounded solutions and the existence of periodic solutions of periodic differential equations, J. Diff. Eq. 160(2000), 263-282.

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Index

AP (X), 28 BC(R, X), 1 BU C(R, X), 16 C0 -semigroup, 7 Λ-class, 93 ε-period, 26 ε-translation, 26 σb (f ), 29 sp(u), 19 spu (f ), 24

condition condition condition condition

H3, H4, H5, H6,

82 183 184 187

domain, 2 eigenspace, 198 evolution semigroup, 63 evolutionary process, 63 exponential dichotomy, 39

accretive, 199 admissible, 83 almost automorphic function, 31 almost periodic function, 26 anti-periodic, 80 approximate eigenvalue, 6 approximate eigenvector, 6 approximate point spectrum, 15 Approximation Theorem, 29 autonomous functional operator, 96

Fourier- Carleman transform, 22 Green function, 42 Higher Order Differential Equations, 89 integer and finite basis, 115 invariant subspace, 197

Beurling spectrum, 19 Bochner’s criterion, 28 Bohr spectrum, 29

linear operator, 2 Lipschitz Inverse Mapping, 193

closed linear operator, 4 compact almost automorphic function, 32 compact semigroup, 12 condition H, 68 condition H1, 82

mild solution of functional evolution equation, 96 mild solution of higher order equations, 90 mild solution on R, 83 mildly admissible, 84 207

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operator LM , 84 orthogonal complement, 198 Perron Theorem, 39 point spectrum, 15 range, 2 relatively dense, 26 residual spectrum, 15 resolvent map, 5

solution on R, 83 spectral inclusion, 14 spectral mapping theorem, 14 spectrum of a function, 23 spectrum of linear operator, 4 strongly continuous group, 16 totally ergodic, 31 trigonometric polynomial, 27 uniform spectrum, 24

self-adjoint, 198 semigroup of type ω, 183, 198

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